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Full text of "An investigation of the effectiveness of existing bridge design methodology in providing adequate structural resistance to seismic disturbances. Phase III : analytical investigations of seismic response of short, single, or multiple-span highway bridges"

gg 2 port No. FHWA-RD-75-10 



.A3 
no. 



i INVESTIGATION OF THE EFFECTIVENESS OF 
fhwa- (ISTING BRIDGE DESIGN METHODOLOGY IN 
?°- PDING ADEQUATE STRUCTURAL RESISTANCE 
; .0 SEISMIC DISTURBANCES. Phase III: Analytical 
Investigations of Seismic Response of Short, Single, 
or Multiple-Span Highway Bridges. — 

M. Chen and J. Penzien jun 2.1976 ' 

I — I 




*Mtes o* 



October 1974 
Final Report 



This document is available to the public 
through the National Technical Information 
Service, Springfield, Virginia 22161 



Prepared for 

FEDERAL HIGHWAY ADMINISTRATION 
Offices of Research & Development 
Washington, DC. 20590 



DISCLAIMER 

The contents of this report reflect the views of the authors who 
are responsible for the facts and the accuracy of the data presented 
herein. The contents do not necessarily reflect the official views 
or policy of the Department of Transportation. This report does not 
constitute a standard, specification or regulation. 



J UN 2.1976 



^ 



Technical Report Documentation Page 



1. Report No. 



FHWA-RD-75-iO , 



2. Government Accession No. 



4. Title and subtiii^/id^invest-jgat-jop £ t ne Effectiveness of 
Existing Bridge Design Methodology in Providing Adequate 
Structural Resistance to Seismic Disturbances. Phase III: 
Analytical Investigations of Seismic Response of Short, 
Single, or Multiple-Span Highway Bridges. • 



3. Recipient's Catalog No. 



5. Report Date 

October 1974 



6. Performing Organization Code 



7. Author's) 

M. Chen and J. Penzien 



8. Performing Organization Report No. 



9. Performing Organization Name and Address 

University of California 
Campus Research Office 
118 California Hall 
Berkeley, California 94720 



10. Work Unit No. (TRAIS) 

35A2-012 



11. Contract or Grant No. 

D0T-FH-1 1-7798 



12. Sponsoring Agency Name and Address 

Office of Research and Development 
Federal Highway Administration , 
ft". S. Department of Transportation 
Washington, D. C. 20590 



13. Type of Report and Period Covered 



Phase III - Final Report 



14. Sponsoring Agency Code 



^fl.^/f 



is. supplementary Notes p^A Contract Manager: J. D. Cooper, HRS-11 
This report is the third in a series. The others in the series are: 
Phase! Report, FHWA-RD-73-13, Literature Survey, NTIS PB 226-718 AS 
Phase II Report. FHWA-RD-74-3 , Long Bridges, NTIS PB 233- 711 AS 

16. Abstract 



I o. MDSiracT 

This report is the third in a series to result from the study, "An Investigation of 
the Effectiveness of Existing Bridge Design Methodology in Providing Adequate Struc- 
tural Resistance to Seismic Disturbances," sponsored by the U. S. Department of 
Transportation, Federal Highway Administration. Descriptions are given to the 
analytical investigations of the seismic response of short, single or multiple-span 
highway bridges of the type where soil -structure interaction effects are important. 

Six different mathematical model elements are incorporated into the computer program 
which possess the capability of performing linear or non-linear analyses. Finite 
element modeling is used for the backfill soils. Bridge deck, piers, and abutments 
are modeled using prismatic beam elements. A frictional element is used to model the 
discontinuous behavior at the interface of backfill soils and abutments. Discontinu- 
ous type expansion joint elements are also included. Linear spring elements provide 
flexibility at the vertical soil boundaries. The soil foundation flexibilities under 
columns are established using elastic half-space theory. In the non-linear mathe- 
matical model the effects of separation and impact at the interface between abutments 
and backfills, the yielding at concrete columns and backfill soils and slippage at 
the expansion joints are taken into consideration. 

Parameter studies are first carried out considering a rigid wall backfill soil system, 
A short, stiff, three-span bridge is then investigated with full soil -structure 
interaction effects included. Finally, based on the analytical results, a general 
conclusion regarding the analyses capability is deduced. 



17. Key Words 

Earthquake, Bridge Seismic Analysis, 
Structural Analysis, Seismic Design 
Criteria. 



18. Distribution Stotement 

No restrictions. This document is avail- 
able through the National Technical Infor- 
mation Service, Springfield, Virginia 
22151 



19. Security Clossif. (of this report) 

Unclassified 



20. Security Classif. (of this page) 

Unclassified 



21. No. of Pages 

176 



22. Price 



Form DOT F 1700.7 (8-72) 



Reproduction of completed page authorized 



FORM FHWA-J21A (REV. 4-*0) 

UNITED STATES GOVERNMENT 

Memorandum 



U.S. DEPARTMENT OF TRANSPORTATION 

FEDERAL HIGHWAY ADMINISTRATION 



FEB 18 1975 



r 

Individual Researchers 



L, 



~1 



DATE: 



In reply refer to: HRS-11 



J 



es u. Coo^vTroject Manager, FCP Project 5A, 
Improved Protection Against Natural Hazards of 
Earthquake and Wind 

subject: Transmittal of Research Report No. FHWA-RD-75-10 

"Effectiveness of Existing Bridge Designs in Resisting Earthquakes, 
Phase III: Short, Single or Multiple-Span Highway Bridges" 

Distributed with this memorandum is the subject report intended 
primarily for research audiences. This report will be of interest 
to structural researchers concerned with earthquake resistant 
highway bridges. 

This recently issued report is the third in a series to result from 
research being conducted at the University of California, Berkeley, 
for the Federal Highway Administration. Analytical investigations of 
the seismic response of short, single or multiple-span highway bridge 
structures of the type which suffered heavy damage during the San 
Fernando earthquake of February 9, 1971, are described in the report. 

Additional copies are available from the National Technical Information 
Service (NTIS), Department of Commerce, 5285 Port Royal Road, Springfield, 
Virginia 22151. A small charge is imposed for copies provided by NTIS. 

Attachment 




iii 
BUY U.S. SAVINGS BONDS REGULARLY ON THE PAYROLL SAVINGS PLAN 



PREFACE 

The investigation with interpretation as described in this 
report was sponsored by the U. S. Department of Transportation, 
Federal Highway Administration, under Contract No. DOT-FH-11-7798 
covering the period July 1, 1971 through September 30, 1974. 

The general investigation called for in this contract is under 
the supervision and technical responsibility of Professors R. W. Clough, 
W. G. Godden, and J. Penzien. Professor Penzien acts as principal 
investigator. 



iv 



ACKNOWLEDGEMENT 

The authors wish to express their sincere appreciation to 
the California State Division of Highways, Department of Public 
Works, for providing the engineering data of the bridge structures 
studied in this investigation. 






v 



TABLE OF CONTENTS 

Page 

FKbr i\v**£i o o • • » • o o o • o • o o o o e o o e ' o o o o • e o o o X V 

ACKNOWLEDGEMENT v 

TABLE OF CONTENTS vi 

LIST OF TABLES . . viii 

LIST OF FIGURES ix 

I. INTRODUCTION 

A.' Statement of Problem 1 

B. Review of Pertinent Investigations .'.......... 1 

C. Review of Damages Caused by Earthquakes ........ 3 

D. Scope of Present Investigation . 4 

II. ELASTIC STIFFNESSES OF MODEL ELEMENTS 

A. Soil Finite Element of a Continuum 7 

B. Soil Boundary Element 14 

C. Prismatic Beam Element 14 

D. Frictional Element 17 

E. Expansion Joint Element .. 19 

F. Equivalent Column for Foundation 19 

III. MATERIAL PROPERTIES OF SOIL 

A. Stress-Strain Relationship 29 

B. Mohr's Envelope, p-q Diagram 32 

C. Active and Passive Stress 34 

D. Damping 36 

IV. NON r LINEAR STIFFNESSES OF MODEL ELEMENTS 

A. General Elastic-Perfectly Plastic Stress-Strain 
Relations 49 

B. Soil Finite Element 52 

C. Column Element 57 

D. Frictional . Element 61 

E. Expansion Joint Element 62 



vi 



TABLE OF CONTENTS (cont.) 

Page 
F. Soil Boundary Element and Equivalent Column of 

Foundation 62 

V." DYNAMIC ANALYSIS PROCEDURES 

A. Equation of Motion 66 

B. Stiffness Matrix ..... 67 

C. Mass Matrix 67 

D. Damping Matrix 68 

E. Step-by-Step Integration Techniques 69 

F. Time Interval At 74 

G. Earthquake Input 75 

VI o PARAMETER STUDIES 

A. Rigid Wall-Backfill System 78 

B. Integration -Time-Interval Sensitivity Analysis 84 

C- Bridge-Soil System 85 

D. Comparison of Resultant Abutment Backfill Force Obtained 

by Analysis and the Monobe-Okabe Method ....... 89 

VII. GENERAL CONCLUSION H8 

BIBLIOGRAPHY 119 

APPENDIX COMPUTER PROGRAM LISTINGS 130 



VI 1 



LIST OF TABLES 



Table- 1 Details of Rigid Wall Systems for Study of Lateral Extent 

Table 2 Comparison of Responses of Rigid Wall System at Different 
Lateral Extent 

Table 3 Comparison of Responses of Rigid Wall System With Different 
Length to Height Ratio R = a/b 

Table 4 Comparison of Response of Rigid Wall System with R = 2 for 
x < 2H and R = 10 for x > 2H at Different Time Step 

Table 5 Comparison of Maximum Total .Seismic Force on the Bridge With or 
Without Soil Interaction 



vm 



LIST OF FIGURES 

Fig. 1 Bridge Structure and Analytical Model 

Fig- 2 Two Dimensional Isoparametric Element 

Fig. 3 Boundary Element 

Fig. 4 Beam Element Coordinate System 

Fig. 5 Frictional Element, Height = 0, Local Coordinate 

Fig. 6 Expansion Joint Element 

Fig. 7 Displacement Types 

Fig. 8 Pier with Four Columns 

Fig. 9 Hysteretic Stress-Strain Relationships at Different Strain 
Amplitudes - Secant Modulus 

Fig. 10 Secant Shear Modulus of Sands at Different Void Ratios 

Fig. 11 'Typical Reduction of Secant Shear Modulus with Shear Strain for 
Saturated Clays 

Fig. 12 Variation of Tangent Modulus with Shear Strain for Sand 

Fig. 13 Idealized Stress-Strain Relationship 

Fig. 14 In-Situ Secant Moduli for Saturated Clays 

Fig. 15 Variation of Tangent Modulus with Shear Strain for Clays 

Fig. 16 Mohr's Envelope for Soil Sample 

Fig. 17 Stresses at Maximum Shear Plane 

Fig. 18 p-q Diagram 

Fig. 19 Mohr Circle 

Fig. 20 Stress Paths for Rankine Active and Passive Conditions 

Fig. 21 Stress Conditions at Field 

Fig. 22 Mohr Circle in 3 Planes 

Fig. 23 Active and Passive Failure 



IX 



LIST OF FIGURES (cont.) 

Fig. 24 Equivalent Viscous Friction vs. Yield 

Fig. 25 Mohr - Coulomb Yield Function 

Fig. 26 Column Interaction Curve 

Fig. 27 Interaction Diagram of Concrete Column 

Fig. 28 Situation at Overshooting 

Fig. 29 Overshooting at Discontinuity 

Fig. 30 Simulated Ground Acceleration Record of the San Fernando Earth- 
quake at the Olive View Hospital Site 

Fig. 31 General Plan of North Connector Undercrossing 

Fig. 32 Abutment and Backfills 

Fig. 33 Rigid Wall System for Studying the Effects of Lateral Extent of 
Backfills 

Fig. 34 Static and Maximum Dynamical Total Lateral Earthpressure for 
Different Lateral Extent L/H 

Fig. 35 Maximum Horizontal Displacement at Points 1 to 6 for Different 
Lateral Extent L/H 

Fig. 36 Maximum Horizontal Accelerations at Points 1 to 6 for Dif- 
ferent Lateral Extent L/H 

Fig. 37 Finite Element Model for Studying the Length to Height Ratio R 

Fig. 38 Static and Maximum Dynamical Total Lateral Pressure for Dif- 
ferent Element Size R = a/b 

Fig. 39 Maximum Horizontal Displacement at Points 1 to 6 for Different 
Element size R = a/b 

Fig. 40 Maximum Horizontal Acceleration at Points 1 to 6 for Different 
Element Size R = a/b 

Fig. 41 Static and Maximum Dynamical Pressure Distribution with Dif- 
ferent Number of Layers - Rigid Wall System 

Fig. 42 Total Lateral Earthpressure - Rigid Wall System 



LIST OF FIGURES Ccont,) 



Fig. 43 Static and Maximum Dynamical Pressure Distribution with Dif- 
ferent Soil Properties *- Rigid Wall System 

Fig. 44 Model for Studying Frictional Element, Rigid Wall System 

Fig. 45 Effect of Stiffness of Frictional Element on Total Lateral 
Force - Rigid Wall System 

Fig. 46 Lateral Force and Horizontal Acceleration at Point No. 1 "- 
Rigid Wall System 

Fig. 47 Vertical Acceleration at Point No. 1 

Fig. 48 North Connector Undercrossing, Bridge-Soil Systems 

Fig. 49 Static and Maximum Dynamical Pressure Distribution with 
Different Boundary Conditions at the Base of Abutment 

Fig. 50 Static and Maximum Dynamical Pressure Distribution of 
Bridge-Soil System with Different Substructures 

Fig. 51 Static and Dynamic Pressure Distribution of Bridge-Soil 
System with Different Degrees of Non-linearity 

Fig. 52 Static and Maximum Dynamical Pressure Distribution of Bridge- 
Soil System with Different Foundation 



xi 



I INTRODUCTION 

A. STATEMENT OF PROBLEM 

The seismic response of shorty single or multiple span, highway 
bridges are greatly affected by the phenomenon of soil-structure inter- 
action. The dynamic forces exerted by backfills on the abutments often 
add significantly to the maximum seismic forces developed in the overall 
structural system. Also, bridges of this type usually have relatively 
short and stiff columns which interact strongly with their supporting 
foundations. Neglecting these interaction effects can lead to large 
errors in predicting design loads. It is therefore the objective of 
this investigation to establish appropriate mathematical models which 
will yield realistic seismic response for certain soil-structure sys- 
tems of this type. Further, computer programs are written to carry out 
time-history dynamic analyses as an aid to the design process. 

B. REVIEW OF PERTINENT RESEARCH INVESTIGATIONS 

Numerous analytical investigations have been carried out in the 
past to determine dynamic earth pressures acting on retaining walls. 
One of the earliest was carried out by Okabe and later a similar study 
was reported by Mononobe and Matsuo [80, 74]*. In each of these inves- 
tigations an equivalent static earth pressure was determined as a 
function of acceleration with its resultant acting on the wall at a 

♦Numbers in square brackets refer to Bibliography numbers. 



-1- 



position corresponding to that of static fluid pressure, i.e. at a height 
one-third the distance from the base. Much later, Valera used the 
finite element method to predict the intensities of seismic earth pres- 
sures acting on rigid walls [106] . In this investigation, nonlinear 
soil behavior was considered. Seed and Whitman summarized 31 investi- 
gations carried out from 1926 to 1969 and made valuable suggestions 
for their application to design [97] . Later, Wood derived analytical 
expressions for earth pressures acting against rigid and rotating walls 
using linear soil properties [117] . All of these investigations were 
concerned with intensities of dynamic pressures but in each case they 
assumed the pressures to act independently of the dynamic response of 
the wall itself. 

In recent years, the influence of soil conditions on the seismic 
response of structural systems has received considerable attention. 
Seed recently published a report covering two main aspects of this prob- 
lem: (1) changes in seismic ground motions adjacent to buildings as a 
result of physical interaction effects, and (2) changes in seismic 
response of buildings as a result of the changes in ground motions due 
to different soil deposits. This investigation did not, however, in- 
clude full dynamic soil-structure interaction effects for structures 
other than buildings [98] . 

Few analytical investigations have been reported in which dynamic 
soil-bridge interaction effects were treated rigorously. One such 
study was reported by Penzien in 1970 [82] . In this particular inves- 
tigation, the soil foundation was represented by a series of lumped 
masses interconnected by bilinear hysteretic shear springs having 
properties which varied with soil depth and also interconnected with 



-2- 



viscous linear dashpots. The bridge deck, supporting piers, and pile 
foundations were also modelled as lumped mass systems. The three dimen- . 
sional effects of the foundation soils were determined using the Mind- 
lin Theory of the elastic half-space. Equations of motion were developed 
which considered all dynamic interaction effects. Recently, Tseng and 
Penzien reported an investigation on the seismic response of long" multi- 
ple span bridges in which the mathematical modelling included soil- 
structure interaction effects at the bases of columns; however, due to 
the long structural types considered, soil-abutment interaction effects 
were neglected [105] . 

As far as experimental results are concerned, most investigations 
have been carried out on simple wall-soil systems. Important contribu- 
tions have been made by Matsuo, Ishii and Arai and Tsuchida, Matsuo 
and Ohara, Tschebotanoff , Ohara, and Niwa 168, 52, 69, 104, 79, 78], 

Shepherd and Charleson, and Shepherd and Sidwell have conducted 
field experiments on an existing bridge in the small loading range 
[100, 101]. Their results show that the soil around the bridge displayed 
high energy absorption qualities, i.e. provided considerable damping to 
the overall soil-structure system. Broms and Ingelson have also con- 
ducted field experiments to determine the static earth pressures acting 
on abutment walls [11] . 

C. REVIEW OF DAMAGES CAUSED BY EARTHQUAKES 

Iwasaki, Penzien, and Clough prepared an extensive summary of the 
damages caused to bridges during earthquakes for the period 1923 to 
1971 [53] . Jennings and Wood, and Lew, Legendecker, and Dikkers have 



-3- 



reported on the damages to bridges during the San Fernando earthquake 
of February 9, 1971 [59, 65] . Duke and Leeds reported similar damages 
caused by the Chilean earthquake of I960, while Rose, Seed, and Migliac- 
cio reported on the Alaskan earthquake 'of 1964 [29, 92] . As indicated 
in these reports, short bridges have suffered damages ranging from 
column failures to cracking, tilting, and even overturning of piers, 
abutments, and wing walls. Clearly, this evidence shows that large 
dynamic forces are induced in the overall structural system by backfill 
earth pressures. 

D. SCOPE OF PRESENT INVESTIGATION 

The present investigation is a study of the seismic response of 
short, single or multiple span, highway bridges having narrow abutment 
backfills as shown in Fig. 1. 

Since the earth pressures acting against the abutments can greatly 
affect the seismic response of the bridge, the mathematical modelling 
includes a two-dimensional soil element representation of the abutment 
backfills. This representation accounts for nonlinear soil properties 
and allows different vertical boundaries to be present as shown in 
Figs. ID, IE and IF. The bridge deck, piers (or columns), and abut- 
ments are modelled using prismatic beam elements which may be permitted 
to have hysteretic yielding properties. A frictional element is used to 
model the nonlinear, discontinuous behavior at the interfaces of back- 
fill soils and abutments and nonlinear, discontinuous type of expansion 
joint element is included. The soil foundation flexibilities under the 
columns are represented by equivalent columns. The mathematical model 



-4- 






of the overall soil-structure system permits the study of interaction 
effects and yields the distribution of forces throughout the system. 
Chapter II of this report describes the different elements used 
in the mathematical modelling and presents the derivations of elastic 
stiffnesses used for these elements. Chapter III discusses soil mate- 
rial properties used in the modelling with particular emphasis on non- 
linear properties. Chapter IV develops the stiffnesses of nonlinear 
elements with emphasis on the derivation of the elasto-plastic force- 
displacement relations for the soil finite elements and the bridge 
column elements. Chapter V presents the numerical techniques used * 
the time-history dynamic analysis while Chapter VI presents the r it 
of parameter studies. Chapter VII presents certain conclusions and re- 
commendations. Finally, listings of the computer program are presented 
in Appendix A. 



-5- 





o 

Q_ 


-I 


X 


Ltl 


CO 


O 


< 


O 


Q 




Z 


-J 


< 


< 


CD 


O 


z 


H 


cr 


>: 



N\V?\Vs\V 



* 
* 
* 



CO 


< 




2 


u: 


< 




Q 




21 




< 


Q 


UJ 


z 


oc 


UJ 


3 


Q 
UJ 


& 


X 


3 


u_ 


tr 


• 


</> 


UJ 






UJ 




O 




o 




oc 




CD 


cr 




UJ 


*■• 


_i 




o 


o 


or 


u. 



fc$$S> 



-6- 



II ELASTIC STIFFNESSES OF MODEL ELEMENTS 

Six basic elements are used in the mathematical modelling of the 
bridge-soil system CD soil finite element of a continuum, (2) soil 
boundary element, C3) frictional element at interface of soil and abut- 
ment wall, (4) prismatic beam element, (5) expansion joint element, 
and (6) equivalent column of foundation. It is the purpose of this 
chapter to describe these elements and to derive their elastic stiff- 
nesses. 

A« SOIL FINITE ELEMENT OF A CONTINUUM 

The bridge-soil system shown in Fig. 1 is subjected to earthquake 
motions in the vertical direction and in one horizontal direction coin- 
ciding with the longitudinal axis of the bridge. Because of the condi- 
tions of symmetry, it is assumed that the soils adjacent to the abut- 
ments respond to these motions in a two-dimensional manner. This 
assumption allows these soils to be modelled using two-dimensional fi- 
nite elements of a continuum interconnected at their nodal points. Ma- 
terial properties are defined individually for each element which may 
have an arbitrary quadrilateral or triangular shape. 

The quadrilateral element used is the isoparametric element for 
which the geometry and displacements are described in terms of the 
same parameters of similar order. Using the natural coordinate system 
and interpolation displacement models, the isoparametric formulation has 
several advantages over the generalized coordinate method [25]. First, 



-7- 



the nodal displacements provide a more direct visualization of actual 
structural deformations than do the generalized displacements. Second, 
it is computationally more efficient since no transformation from one 
coordinate system to another is required. Finally, the local coordinate 
system of the isoparametric approach coincides with the- global coordi- 
nate system; thus, eliminating the need for transformation of loads and 
stiffnesses from one coordinate frame to the other. 

The following derivation of the stiffness matrix for the two- 
dimensional quadrilateral isoparametric element is similar to that found 
in certain textbooks 125, 34, 118] . 

1. Coordinate System - The global Cartesian coordinates (x,y) and 

the local (or natural) coordinates (s,t) as shown in Fig. 2 are related 

through an interpolation function h. by the relations 

k 
x = E h. x. 

k 

y = E h. y. 

where 



h = H (1-s) (1-t) 

l 

h. - h (1+s) (1-t) 
2 (2) 

h - h (1+s) (1+t) 

3 

h = h (1-s) (1+t) 



2. Strain-Displacement Equations - The displacements are approxi- 
mated using the same interpolation function which can provide for both 



-8- 



flexible and rigid body modes. In this case, one can state 



u (s f t) = .£, h. u 
x 1=1 i x 



u (s,t) = .£., h. u 
Y i=l i Y L 



(3) 



The two-dimensional strains are obtained by taking derivations of the 
displacements with respect to x and y in the following manner: 



xx 



e = /g 



yy 



xy 




(4) 



3h. 
L , 1 i ' 

1=1 dy x. dx y, 



Since functions h. (i=l,2,3,4) are expressed in terms of s and t, the 
chain rule of derivations can be applied using the equations 



oh. 

1 

8x 


ah. 
i 

a s 


a ah. 

OS 1 

a x at 


3t 
3x 


oh. 

_ l 


3h. 

l 


ah. 

OS _1_ 


3t 



3y 



3s 



3y 3t 



ay 



(5) 



The chain rule is 



3( )■ ) 
3s ( 


3x 
3s 


3s 


\ 3( ) 
J 3x 


3( ) f 

at 


3x 
3t 


3y_ 
3t 


) 3( ) 
( 3y 



(6) 



-9- 



which in its inverted form becomes 



3( ) 



3x 
3C ) 





3v. 

3t 


3s 


f 3( ) 
\ 3s 


1 






) 


J 


3x 
~ 3t 


3x 
3s 


) 3< ) 
/ 3t 



3y 



where the Jacobian J is defined by 



j = a* h. _ ?JL h. 
3s 3t 3t 3s 



Thus, the required derivatives are given by 



3s 
3x 


3t 
3x 


1^ 


3y_ 

3t 


-3v_ 
3s 


3s 
3y 


3t 
3y 


J 


-3x 
3t 


3x 
3s 



Taking derivatives of Eqs. (1) , one can write 

3h. 



3x 
3s " 


^ i 
£ — *— x. 

ds l 
1=1 


3x 
3t " 


f 3h. 

£ -57- x - 
3t l 
1=1 


3s 


«♦ 3h. 

3s i 



1=1 



3 «t 3h. 
^ = I 



3t 



1=1 



3t *i 



(7) 



(8) 



(9) 



(10) 



Making use of Eqs. (2) and (10), the Jacobian, as defined by Eq. (8) 
becomes 



-10- 



J = 8 tS(X 3 - V (y i - Y 2 ] ' (X 1 ~ V (y 3 7!V. 

+ t(x 2 - x 3 ) (y 2 - y.) - (x x - xj (y 2 - y^ (11) 

+ (x x - x 3 ) (y 2 - yj - (x 2 - Xlf ) (y x - y 3 )] - 

and the strain vector defined by Eq. (4) becomes 



£ = 



3n. 
l 

3x 





3h 

2 

3x 





3h 

3 

3x 





3h 
<3x 








3h 
i 

3y 





3h 

2 

3y 





3h 

3 

3y 





3h 
3y 


3h 
l 


3h 
l 


3h 

2 


3h 

2 


3h 

3 


3h 

3 


3h 


3h 


3y 


3x 


17 


3x 


3y 


3x 


3y 


3x 



or in compact form may be expressed as 



IX 



u 



l Yi 



u 



2X 



U 



2 Y 



u 



3X 



U 



3y 

i 
^x 

1 



(12) 



e = B u 



(13) 



The derivatives in the coefficient matrix of Eq. (12) can now be 
expressed in their final maniputable' form 



3h 
l 

3x 
3h 

2 

3x 



3h 



3x 

3h 
«»_ 

3x 

3h 
l 

3y 



3 _ 



~- I y (h + h) +y (h-h) - y (h + h)] 

16J 212 3 «f 2 «f 1 h 

r^- [-y (h + h ) + y (h + h ) + y (h - - h ) ] 

lbJ 112 323 <fl3 



r^-r [y (h-h) -y (h + h) +y (h + h)] 

16J 12 h 22 3«»3 «f 



y— [y (h + h) +y (h-h) - y (h + h )] 

16J 11 i» 23 1 3 «♦ 3 

t^t [-x (h + h ) + x (h - h ) + x (h + h ) ] 

16J 212 3 2 «f «♦ 1 «t 



-11- 



8h 

-57- = TTT I x (h + h) -x (h + h) + x (h-h)] 

dy 16J 112 323 4 3 1 

-T 1 - TZT I x Ch - h j + x (h + h ) - x (h + h )] 

dy 16J 1 if 2 2 2 3 4 3 4 

3h 

-5^- = — - [-x (h + h ) + x (h - h ) + x (h + h )] 

dy 16J 114 213 334 

(14) 

Using standard index notation with i permutating 1 through 4, Eqs- (14) 
can be written in the two single equations 



dh. 

-£■ = tIt I y-^ < h - + h - A ) + y-j. Ch.. - h. A ) - y. A (h.+ h. A 1] 

dX 16J 'l+l 1 1+1 1+2 1+3 1+1 1+3 1 1+3 

3h. , 

-T = TFT I-y-j. < h - + h -u. > + *- A 0». A - h_ ) + x_ (h.+ h.^ )] 

dy 16J ■'l+l 1 1+1 1+2 1+1 1+3 1+3 1 1+3 

(15) 



3. Element Stiffness and Numerical Integration - Considering a 
thickness 03 (normal to the plane of element shown in Fig. 2) , the ele- 
ment stiffness matrix can be expressed in the form 

K = / B_ T C_ B dv / B_ T C_ B_ OJdA (16) 

vol area 

where C_ is the stress-strain matrix with integration being carried out 
over the entire area of the element. For purposes of numerical integra- 
tion, Eq. (16) can be written in terms of the s and t coordinates giving 

1 1 T 
K = / / B C B (det J) Uids dt (17) . 

-l ^l 
Upon application of standard one-dimensional numerical integration 
formulas, Eq. (17) becomes 



-12- 



K = E E a), a), (det J) B (s lf t) CB (s. f t)w (18) 
- - k ] k - 3 k 3 k 

in which s . and t are integration" points and o) . and to are the appro- 
priate weighting functions. Using the Gauss-Legendre quadrature formula 
of degree 2, one obtains 



s. = + 0.577350 
l — 

t. = + 0.577350 

l — 

03. = 1 

1 



i = 1,2 



(19) 



For plane stress, matrix C has the form 



£=r^ 



1 V 

V 





1 

l-v 



(20) 



while for plane strain it becomes 



- " (1+V) (1-2V) 



1-V V 
V 1-V 





1-2V 



(21) 



where E is Young's Modulus and V is Poisson's ratio. 

As shown above, the final form of the element stiffness matrix is 
a function of geometry, element thickness, and material properties. 
The overall stiffness matrix of the entire system is assembled by the 
direct stiffness method [22] . The band width of this matrix depends 
upon the system of numbering nodal points. Therefore, the nodal points 
should be numbered in a manner to minimize computer storage requirement 



-13- 



and operating time. 

B. SOIL BOUNDARY ELEMENT 

Boundary elements are used in the mathematical model to account 
for the elastic action which occurs at the vertical boundaries of the 
soils being considered. The type of element used for this purpose is a 
simple elastic spring as shown in Fig. 3„ This element has a stiffness 
matrix k in the local coordinate system of the form 



k = E 



-1 



-1 



(22) 



where E is the equivalent stiffness of the spring, which can be obtained 
using standard methods [82,89] . 

C. PRISMATIC BEAM ELEMENT 

Prismatic beam elements are used to model the bridge deck, piers, 
abutment walls and equivalent columns for soil foundations. The deri- 
vation of the stiffness matrix for a beam element considering axial , 
shear, and bending deformations can be found in many textbooks [86] . 
Therefore, only the main features of this matrix will be presented here. 

The force components acting on the beam element are axial forces 
s and s , shearing forces s and s , and bending moments s and s as 

1 H 2 5 3d 

shown in Fig. 4. The positive directions of these force components (s) 
and their respective displacement components (u 1 ) correspond to those 
shown in the figure. The stiffness matrix for this element in terms of 



-14- 



the local coordinates is given by the standard form 



—(6X6) 



EA 
£ 



12EI 



A 3 (l+cf ) 



6EI 



z 2 a+i ) 
y 



EA 
£ 



-12EI 



£ 3 d4 ) 
y 



6EI 



£ 2 u4 ) 

y 



(4+i )EI 
T y z 

£u4 ) 
y 



-6EI 



£ 2 (1+<|> ) 

y 

(2-d )EI 

Y S 

A (1+4 ) 

y 



SYMMETRICAL 



EA 



12EI 



£ 3 (1++ ) 

y 



-6EI 



£ 2 (l+<!> ) 

y 



(4+<j> )EI 

y ; 

£(14 ) 
y 



where 



12EI 



G Asy £' 



24U+V) ~- (-4) 
Asy £ 



C23) 



(24) 



represents the shear deformation parameter for reinforced concrete 
elements and where 



£ = 



z 

A = 

Asy = 



element length 

radius of gyration about z axis 

Moment of inertia about z axis 

cross sectional area 

equivalent shear area in y direction 



-15- 



E = modulus of elasticity 

G = modulus of elasticity in shear 

V = Poisson's ratio 

The matrix equation relating displacements in the local coordinates 
u* to displacements in the global coordinates u_ is given by 



u' 



X u 



(25) 



where X has the two-dimensional form 



X = 



^l 


y* 














V* 


D x /D * 




















1 




















V** 


y I 














-y * 


y * 




















l 



(26) 



and where 



D = 



D = 



D„ = 



x. 

D 


- x. 

1 




y j 


- y i 




^D 


2 + D 


2 



(27) 



Thus, the stiffness matrix K in global coordinates becomes 
K = X T k X 



(28) 



as indicated above, the stiffness matrix is linearly proportional to 
the areas and moments of inertia. Therefore, a unit width analysis can 
be carried out, if desired, by dividing all areas and moments of 



-16- 



inertia by the width of the bridge deck. 

D. FRICTIONAL ELEMENT 

A so called frictional element is used to model the frictional 

action, separation, and impact which take place at the interfaces of 

* 

soil backfills and abutment walls. , This element has the following 

characteristics (1) the frictional force per unit area is proportional 

to the normal interface pressure and a coefficient of friction; thus, 

slippage occurs when the direction angle of the resultant of pressure 

and friction exceeds the soil angle of friction, (2) impact occurs at 

the interface upon closure of any gap which may have earlier developed, 

and (3) no frictional resistance can develop at the interface when wall 

and soil surfaces have separated. Discontinuous elements similar to 

this have been developed by Ghaboussi and Wilson, Scholes and Strover, 

White and Enderly, and Tseng and Penzien [40, 94, 111, 105]; however, 

the element developed by Goodman and Taylor has been adopted here [41] . 

This frictional element is a four nodal element as shown in Fig. 5 
having length L but with a height equal to zero, i.e. nodal points 1 
and 4 coincide as do points 2 and 3. It is shown in a local coordinate 
system with the origin at the center of the element and the x axis di- 
rected along the length of the element. 

The relative displacement vector u_ is expressed in terms of the 
displacement vector u. through the linear interpolation function for- 
mula 



-17- 



u = 



, top bottom, 
(u * - u ) 

X. X 

top bottom, 
(u * - u ) 



or 



u = 



(1- i*-) -(1+ |*-) Cl+~) (1-^-) 
-(1-1*3 o -U+~* (1+|^ (1+7^ 

Li Jj Jj Lt 



r U 



IX 



u 



u 



fu 



i f 

2X 
t \ 

2y 



u 



3Xi 



u 



3y 



u 



i»X 



u 



«»y 



(29) 



The material property matrix k_ expressing the stiffness per 
unit length in the normal and tangential directions is given by 



k = 



k 

s 



n 



(30) 



Upon applying the variational principle of solid mechanics, the elastic 
stiffness matrix in the local coordinate system becomes 



K = 



2k 

s 





Ik 

s 





-Ik 

s 





-2k 

s 








2k 
n 





Ik 
n 





~ lk n 





-2k 

] 


Ik 

s 





2k 

s 





-2k 

s 





-Ik 

s 








Ik 
n 





2k 
n 





-2k 
n 





-Ik 


-Ik 
s 





-2k 

s 





2k 

s 





Ik 

s 








-Ik 
n 





-2k 
n 





2k 
n 





Ik 


" 2k s 





-Ik 

s 





Ik 

s 





2k 

s 








-2k 





-Ik 





Ik 





2k 



n 



n 



n 



n 



n 



n 



n 



(31) 



-18- 



Ec EXPANSION JOINT ELEMENT 

An expansion joint may be present between the bridge deck and an 
abutment as shown in Fig. 6. This joint can develop horizontal fric- 
tional forces which should be modelled properly. For this purpose, 
the frictional element previously described can be adopted; however, 
two additional boundary conditions must be imposed, namely, (1) the 
relative displacement of the upper and lower part of the joint element, 
u - u , cannot cause a gap closure (between deck and abutment) great- 

3X 2X - 

er than the original gap dimension, and (2) frictional forces can be 
developed only when the relative displacement, u - u , causing a 

IX 3 X 

widening of the gap is less than the original "seat" dimension. As soon 
as the relative displacement (u - u ) exceeds the original seat di- 
mension, the bridge deck falls from its support; thus, the computer 
analysis is stopped at this point. 



F. EQUIVALENT COLUMN FOR FOUNDATION 

Various mathematical models have been used for structural founda- 
tions. As reported in the literature, Parmelee, Whitman and Roesset, 
Dobry, and others [81, 108] used a simple spring dashpot model; Jennings 
and Bielak and Richart, Hall, and Woods [58, 89] used the elastic half 
space; Whitman [109] used an equivalent lumped mass model; and, Dans 
and Butterfield, Finn, Lysmer and Kuhlemeyer, and Wilson {31, 36, 66, 
114] used a finite element mesh. Various methods have been reported 
for estimating the lateral stiffness of foundations employing piles 
[84, 88] . 



-19- 



The replacement of the foundation flexibility with an equivalent 
column through use of the elastic half space theory and under the 
assumption of quasi-static behavior has been reported by Penzien, et. 
al. [8] . This method has been adopted in the present investigations. 

The first step in finding the equivalent column for the founda- 
tion is to determine the lateral, vertical, and rotational stiffnesses 
of the foundation at the footing (or pile cap) level. Once these three 
stiffnesses have been determined, the foundation is replaced by a column 
of length L, flexural stiffness EI, and axial stiffness AE which when 
fixed at its base provides the equivalent lateral, vertical, and ro- 
tational stiffnesses to the footing. The stiffness matrix for this 
column is given in the form of Eq. (23) under the assumption of no 
shear deformation, i.e. <J> =0. 

y 

The foundation stiffnesses can be obtained by either the numerical 
procedure outlined by Penzien or by a closed form approach reported by 
Gerrand and Harrison [82, 39] . 

1. Closed Form Solution fpr Lateral Stiffness of Circular Foot- 
ing - The closed form solution for lateral stiffness is available for 
a circular shaped footing as shown in Fig. 7-c. If the footing is 
rectangular in shape, an equivalent radius r should be calculated for 
a circular footing having the same area, then 
8 r G 

\ ■ sk- (32 > 

where 

k = lateral stiffness for a single footing 
r » radius of footing 

-20- 



G = shear modulus of soil 
V = Poisson's ratio 

For a pier of multiple supports, the interactions between footings are 
estimated by Eq. (33) which describes the lateral displacement at a 
distance r from a loaded' footing. 



1 r,„ .., .-11. „ (r 2 -l) 



u = T >, TZ \ t(2-V)-sin - + V - ■ > ' ) C33) 

h 4Trr Q G r r 2 

where 

u = the displacement 

T, = the total applied force at a single footing 
n 

r = the distance in terms of radius r 

o 

By calculating the average displacement of all footings in a pier and 

dividing the total force by the average displacement, the equivalent 

stiffness of a pier with multiple, footings is then obtained. 



2. Numerical Solution Using the Mindlin Equation to Calculate 
Lateral Stiffness - The Mindlin equation has the following form. 
When the x component of displacement on the surface is to be calculated, 
as produced by a single concentrated force P located at the origin 
(o, o, o) on the surface of an isotropic half space and acting in the 
x-direction 



u (x , y , o) . Pte,o.o) U-M + L + 4g-v)(i-2v) 

x '*' 16lTCl-V)G ] R R R 

+ x 2 [ I + 3 ~ 4v _ 4 (1-V) (1-2V) 
R 3 R 3 R 3 



-21- 



(34) 



in which 

R 2 = x 2 + y 2 

Under the assumption that the lateral force is uniformly distributed 
over the area of the footing, the procedure to calculate the lateral 
stiffness of a single footing is (1) replace the uniform pressure by 
as many concentrated loads P as accuracy requires, (2) calculate the 
lateral displacement of each load point over the footing area due to 
each concentrated load, (3) sum the resulting displacements at each 
load point within the footing as caused by the full set of loads, (4) 
average the load point displacements, and (5) divide the total resultant 
force by the average of the load point displacements to obtain the 
lateral stiffness. 

Again, if a pier has many footings, the same averaging procedure 
is applied by assuming a concentrated force at the center of each foot- 
ing; thus, the lateral stiffness of each pier foundation can be obtained. 

3. Comparison of Two Methods for Lateral Stiffness - To compare 
the two previously described methods for obtaining foundation lateral 
stiffness, consider the pier shown in Pig. 8 having four footings. 
Assuming the footing is 8.5 x 8.5 ft and the distance between footings 
is 26 ft, the individual footing stiffness k and pier stiffness k 
obtained by the two approaches have the following values in k/ft: 

!32.5 G Numerical method 
25.5 G closed form solution 

f 100.6 G Numerical method 

P 1 

/ 80.0 G closed form solution 



-22- 



2 

Where the unit of G is in k/ft . 

In each case the stiffnesses differ by approximately 20%. In the 

numerical solution the footing is considered to have 36 concentrated 

forces applied at equal grid intervals over the area. If the resultant 

force had been discritized at more than 36 points, the differences in 

» 
stiffness would have decreased. 

4. Vertical Stiffness with Friction Piles - Assuming the friction 
force per unit length of pile varies in a linear manner from a maximum 
value at the top to zero at the bottom, and assuming zero vertical dis- 
placement at the bottom of the pile, the vertical stiffness of the pile 



k is 
v 



k = 3AE/L (35) 



Where L, A, and E are the length, area, and modulus of elasticity, 
respectively, of the pile. 

5. Vertical Stiffness Without Piles - Assuming uniform vertical 
displacements as shown in Fig. 7-a, the vertical stiffness of a circular 
footing without piles is given by 

4 r G 

k = . ° (36) 

v 1-v 

For other footing shapes, similar stiffness relations have been reported 
by Lysrrer and Duncan [67] . 

6. Rotational Stiffness Without Piles - Assuming rotational 
displacements as shown in Fig. 7-b, the rotational stiffness of a cir- 
cular footing without piles is given by 



-23- 



8 r 3 G 



3(1-V) 



(37) 



The rotational stiffness of a rigid footing with piles can be calculated 
directly from the vertical stiffness of each pile. 



-24- 



Y 

> i 



3(X 3 ,Y 3 ) 




2(X 2l Y 2 ) 



KX,,Y,) 



3 ( UO 




2(1,-1) 



(-1,-1) 



a. GLOBAL COORDINATE 



»-X 

b. LOCAL COORDINATE 



FIG. 2 TWO DIMENSIONAL ISOPARAMETRIC ELEMENT 



v> 



jS*^ 



s 2 



LOCAL COORDINATE 



-^"X 



GLOBAL COORDINATE 



FIG. 3 BOUNDARY ELEMENT 



-25- 




^x 



GLOBAL COORDINATE 



FIG. 4 BEAM ELEMENT COORDINATE SYSTEM 



y 
A 



TOP 



Y=0 



BOTTOM Y = 



L/2 



+ 



J 3y 



L/2 



-^-x 



♦•U3X 



->' 



FIG. 5 FRICTIONAL ELEMENT, HEIGHT = 
LOCAL COORDINATE 



-26- 



UPPER PART 



t 



3 

rTTfflTTI 



GAP 



4- 



SEAT 



N 

LOWER PART 

a. PROTOTYPE 



UPPER PART ^-«- 



3 ' 
2 



b. MODEL 



LOWER PART 



FIG. 6 EXPANSION JOINT ELEMENT 



-27- 




(a) UNIFORM VERTICAL DISPLACEMENT 




(b) ROTATIONAL DISPLACEMENT 




(c) LATERAL DISPLACEMENT 



3 AT 26' 



8.5' 



8.5 



^> 



FIG. 7 DISPLACEMENT TYPES FIG. 8 PIER WITH FOUR 

COLUMNS 



-28- 



Ill MATERIAL PROPERTIES OF SOIL 

In this chapter, the pertinent soil material properties used to 
establish a non-linear finite element model are discussed. The non- 
linear stress-strain relations, Mohr's envelope and p-q diagram, the 
concepts of active and passive stresses, and the damping characteristics 
of soil are described. 

A. STRESS-STRAIN RELATIONSHIP 

Basic to establishing the force-displacement relationships for 
soil elements as employed in the mathematical model are the stress- 
strain relationships for the materials involved. These materials in- 
clude both cohesionless and cohesive soils. 

Numerous authors, including Bishop and Henkel, Bishop, Comforth 
and Seed, have reported procedures for determining the shear modulus of 
sand [9, 10, 23, 96] . The investigators show that the shear modulus 
is strongly influenced by confining pressure, strain amplitude, and 
void ratio. In the present investigation, the equivalent secant shear 
modulus as determined by extreme points on the hysteresis loop, Fig. 9, 
is adopted. This modulus can be estimated using the relation proposed 
by Seed [96] , namely, 

. \ 
G = 1000 k (O* ) psf (38) 

2 m c 

where G is the secant modulus of sand, 0* is the effective mean stress 

m 

and k is a parameter which depends upon void ratio e and strain amplitude 

-29- 



as shown in Fig. 10. Triaxial tests show that k depends only upon 

2 

void ratio e at very low strain levels (y ^ 10 ' percent) . At inter- 
mediate strain levels (10 <^ y <_ 10 percent) , it still depends pri- 
marily upon void ratio but is also slightly influenced by state of 
stress and the friction angle <}> . At very high strain levels, k is 

2 

essentially a constant which is almost independent of state of stress, 
friction angle, and the void ratio. Thus for practical purposes, the 
values of k can be assumed to vary only with strain amplitude and void 
ratio as shown in Fig. 10. 

For prescribed values of void ratio 'and confining pressure, the 
stress-strain relationship is non-linear with the shear modulus chang- 
ing with shear strain. Upon the initiation of yielding the modulus 
becomes very small. A tangent modulus curve transformed from the equi- 
valent shear modulus curve, Fig. 10, is shown in Fig. 12. Theoretically, 
one could use a revised tangent modulus over each time interval of a 
dynamic analysis; however, it is believed to be more practical to use 
a more simplified form of stress-strain relationship. In the present 
investigation a trilinear stress-strain relationship, Fig. 13-a,has been 
adopted with the shear modulus remaining constant during each of three dif- 
ferent loading stages. In the initial stage, the soil element is in 
its geostatic state, i.e. the vertical stress is equal to the weight 
of- soil (per unit area) above the point of interest as given by 

y 

<J = / w dy (39) 

o 

where y is the depth and w is the unit weight of soil. The lateral 
stress in the principal horizontal direction is 



-30- 



a = k a (40.) 

h o v 

while in the orthogonal horizontal direction (z) the stress O for 
plane stress is 



= (41) 

z 

and for plane strain is 

k 

a = u(a + a. ) = . ° (a +ka) = k a = ex (42) 
z v h l+k v o v o v h 

o 

where ]i is Poisson's ratio. The shear modulus in this stage is evalua- 
ted by Eq. (38) with k selected in accordance with the very low strain 
level shown in Fig. 10, In the second stage the soils in the backfill 
are no longer in the geostatic state due to loadings from the bridge. 
In this case the shear modulus is calculated using Eq. (38) , but k 
is revised to be consistent with the maximum shear strain in the ele- 
ment. The third stage occurs after the initiation of yielding in which 
case the stress-strain relation is evaluated in accordance with a theory 
of plasticity as described in Chapter IV. 

In those cases where curves of k vs. Y as shown in Fig. 10 are 

2 

unavailable, a bilinear stress-strain relationship is adopted with 
the initial modulus being estimated by an empirical method and the 
second stage modulus being calculated by the theory of plasticity as 
shown in Fig. 13-b. 

Turning our attention now to certain clay materials, test data 
show that at very low strain levels, the shear modulus varies almost 
linearly with shear strength [112] . In a summary report, Seed has 
presented a curve of secant shear modulus versus shear strain for 

-31- 



saturated clays {96] . The shear modulus expressed by this curve, Fig. 

14 , is in the normalized form G/s where s is the undrained shear 

u u 

strength of the material. A curve showing the degradation of secant 
shear modulus with shear strain, as obtained by a number of investiga- 
tors, is shown in Fig. 11 [96]. These relationships can be used as a 
guide for estimating the initial modulus of clay when laboratory test 
data on the specific material are unavailable. 

Fig. 15 gives a tangent modulus curve for clay which is consistent 
with the secant modulus curve of Fig. 11. Test results show that the 
shear modulus is essentially independent of the confining pressure. 
A re-evaluation of shear modulus before the initiation of yielding be- 
comes much less important for this material than for sand. For this 
reason, a bilinear stress-strain relation has been assumed for cohesive 
soils in the present investigation. This initial modulus is estimated 
from the data previously described and the tangent stiffness after 
yielding is obtained using the same basic procedure as used for sands. 
No hardening effects after the initiation of yielding are considered. 

B. MOHR'S ENVELOPE, p-q DIAGRAM 

The most widely used yield criterion in soil mechanics is the 
Mohr -Coulomb criterion which relates the normal stress O and the 
shear stress T at the failure plane by the equation 

X = c + tan <)> (43) 

where c is the cohesion of the soil and ty is its friction angle. The 
soil coefficients are usually obtained by conducting triaxial tests at 



-32- 



various confining pressures and by drawing a common tangent line to 

the resulting Mohr's circles as shown in Fig. 16. it should be noted 

that since the confining pressure -of the triaxial test is uniform, 

i.e. = , the Mohr's circle can be used in its familiar two- 
2 3 

dimensional form. Equation (43) when plotted on the O-T plane is called 
Mohr's yield envelope whidh implies (1) elastic behavior for a state of 
stress whose Mohr's circle lies entirely below the envelope, (2) yield- 
ing, or impending yielding, for a state of stress whose Mohr's circle 
has the envelope as its tangent, and (3) that any state of stress whose 
Mohr's circle crosses the envelope "is not permitted. 

An alternate way of plotting the results of triaxial tests is to 
plot the stresses at the plane of maximum shear on the p-q plane, 
namely, 

a + a 

p = 1 = 3 (44) 



and 



a - a 

q .= - n 3 (45) 



In Fig. 17, points A, B, and C in the 0-T plane represent the stresses 
on planes of maximum shear. A line drawn through these same points 
in the p-q plane, as shown in Fig. 18, is called the k line. The 
equation of this line is 

q = a + p tan a (46) 

Strength parameters c and <f) can be computed from Eq. (46) through 
the relations 

sin (f> = tan a (47) 



-33- 



c = a/cos <f> (48) 

Co ACTIVE AND PASSIVE STRESS 

Using the previously defined yield criterion, active and passive 
soil stresses can be defined as they relate to the backfill pressures 
exerted on abutment walls. Starting with soil equilibrium in its 
geostatic state which has principal axes in the vertical and horizontal 
directions, decrease the horizontal compressive stress continually until 
the shear strength of the soil is reached and failure occurs. The hori- 
zontal compressive stress at the point of failure is active stress. On 
the other hand, if the horizontal compressive stress is continually 
increased, the shear strength of the soil will again be reached at a 
much higher stress level called the passive stress. The Mohr's circles 
representing these two failure conditions are shown in Fig. 19 where 
G and a represent the vertical and horizontal stresses, respectively, 
in the geostatic state arid Q and represent the active and passive 
stresses, respectively. It should be noted here that since the two 
horizontal stresses are assumed equal, only two stress components are 
required to describe the stress conditions. 

The stress conditions in the C-T diagram of Fig. 19 are again 
shown in the p-q diagram of Fig. 20, where point A is the geostatic 
state of stress, point B is the passive state of stress, and point C 
is the active state of stress. Lines AC and AB are stress paths which 
depict the successive states of stress which occur in changing from 
the geostatic condition to the active and passive conditions, respective- 
ly. 

-34- 



When applying the concepts of active and passive stresses to more 
complex stress conditions, certain modifications are needed. For ex- 
ample, the stress conditions of the bridge backfill-soil "treated in this 
investigation is essentially three-dimensional. In this case shear 
stresses may exist on both vertical and horizontal planes even under 
gravity loadings. Referring to Figs. 1-A and B for the element coordi- 
nates, the Mohr's circle for this condition may be as shown in Fig. 21. 
Further, it should be noted that if the side slopes of the embankment 
as seen in Fig. 1-B are not sufficiently flat, the yield plane will 
show as a line in the y-z plane. In the present study however it is 
assumed that these slopes are sufficiently flat so that the yield plane 
shows as a line in the x-y plane. Thus the state of stress at failure 
as shown on an element of soil in the x-y plane, can be represented by 
a Mohr's circle tangent to the Mohr's envelope as shown by Circle 1 
in Fig. 22. States of stress in the x-z and y-z planes would be repre- 
sented by Mohr's circles falling entirely within the above defined cir- 
cle as shown by Circle II and III in Fig. 22. Because of these restric- 
tions, the problem can be treated as a two-dimensional problem in the 
x-y plane. Finally, in the present investigation, all stresses vary 
prior to reaching a state of yield. Based on the previous assumptions, 
active and passive states can be defined in an equivalent manner. 
Referring to Fig. 23 where the absolute values of the p and q stresses 
are plotted, p values at static loading and failure conditions can be 
compared. If |p[ is assumed to decrease continually with increasing 
|q| stress until the Mohr ' s envelope is reached, active failure occurs. 
On the other hand, if [p( is assumed to increase continually with in- 
creasing |q| stress until the envelope is reached, passive failure 

-35- 



occurs. Stresses p and p represent the active and passive stresses 
in this case. 



Do DAMPING 

Soils, like all materials, exhibit energy dissipation when subjected 
to cyclic loading. Different methods have been used for measuring damp- 
ing depending upon the strain levels involved [96] t . Forced vibration 

_2 

tests have been used for strain levels in the range 10 to 10 percent, 
free vibration tests have been used for Strain levels in the range 10 3 
to 1.0 percent, and static tests have been used to measure the hystere— 
tic energy absorption for strain levels in the range 10 ' ; to 5 percent. 
The energy dissipation in the latter case has been expressed in terms 
of equivalent viscous damping ratios [102, 103] . 

Experimental evidence shows that two major factors influence the 
amount of damping exhibited by sands, namely, confining pressure and 
strain amplitude. Damping in this case tends to decrease with an in- 
crease in confining pressure and tends to increase with strain ampli- 
tude. Clay materials, on the other hand, exhibit damping which is 
essentially independent of confining pressure but, like sands, the 
damping increases with strain amplitude. 

In modelling the damping as measured in soils, it has generally 
been carried out by using an equivalent viscous damping system. An 
important basic study carried out along these lines, using a wide 
range of yield values, has been reported by Hudson 147]; see Fig. 24. 
In this case, the equivalent viscous damping ratio corresponding to 
elasto-plastic hysteretic damping for a single degree of freedom system 

-36- 



is given. 

In the present investigation, hysteretic damping has not been 
converted to an equivalent viscous' form. Rather, it is treated in its 
true strain dependent form. Since hysteretic damping treated in this 
manner accounts for energy dissipation only for strain levels above 
yield, it is necessary to also include viscous damping to represent the 
low strain level velocity dependent damping. For this purpose, Rayleigh 
damping has been used in the present investigation. 



-37- 




^Strain 



FIG. 9 HYSTERETIC STRESS -STRAIN RELATIONSHIPS AT 
DIFFERENT STRAIN AMPLITUDES -SECANT MODULUS 



-38- 







io- 2 

Shear Strain -percent 
FIG. 10 SECANT SHEAR MODULI OF SANDS AT DIFFERENT VOID RATIOS 

(After Seed and Idriss) 



1.2 



1.0 





c 




K 






C 


k. 






«D 




o 


a. 


OR 




<f 




in 


■ 
O 










o 


M 




a> 


to 






« 


0.6 








o 






V) 


o 

■-'-— 




3 












o 


3 


0.4 


2 


•o 
o 




i- 


? 




o 






.c 
C/l 


s 


0.2 



c/> 




10 



FI6.II 



IO" 2 10"' 1 

Shear Strain, y - percent 

TYPICAL REDUCTION OF SECANT SHEAR MODULUS 
WITH SHEAR STRAIN FOR SATURATED CLAYS 

(After Seed and Idriss) 



10 



-39- 



< 
q: 




< 
CO 

01 
O 



< 

H 
CO 

< 
u 

CO 

X 



CO 

-J 

Q 
O 



O 



01 



CNJ 
Ll. 



-40- 



-YIELD STRESS 



STATIC- 
STRESS 




-»-€ 



(a) TRILINEAR CURVE 



a 



YIELD - 
STRESS 



7 



■*-€ 



(b) BILINEAR CURVE 



FIG. 13 IDEALIZED STRESS -STRAIN RELATIONSHIP 



-41- 




CO 

5 



Q 
Ui 

a 

CO 

QC 
O 



_l 


<o 


3 
O 




O 


TJ 


s 


~ ™* 




■o 


01 
< 
UJ 


c 
o 


X 


-o 


CO 


0) 




0> 


-3» 


CO 



< V. 

o a> 

m *- 

co h- 

3 5 

CO 

i 
z 



e> 



-42- 



< 

\- 
co 



X' 



% 

_l 
o 

a: 

2 

< 

tr 
I- 
co 

a: 
< 
in 

x 

CO 



CO 

-J 

Q 
O 



\ 



\ 



co 

CO 
UJ 

a:* 
co 



2<=J 



UJ 



< 

% 

u. 



-43- 



A 



MOHR'S ENVELOPE 




FIG. 16 MOHR'S ENVELOP FOR SOIL SAMPLE 



A 



MOHR'S ENVELOPE 




COMPRESSION 



FIG. 17 STRESSES AT MAXIMUM SHEAR PLANE 



^44- 



q = 



°l-°*3 



1 


2 




K f -LINE 






A^ 


B ^^a 


a 

1 


±^* 




— q =a+p tana 



FIG. 18 p-q DIAGRAM 



FIG. 19 MOHR CIRCLE 



P = 



o-, +cr 3 




•<X 



-45- 



K f -LINE 




K f -LINE 



FIG. 20 STRESS PATHS FOR RANKINE ACTIVE 
AND PASSIVE CONDITIONS 



-a»Z 




(°z,Tyz> 




(cr y ,T xy ) 
(Oy.fyz) 



FIG. 21 STRESS CONDITIONS AT FIELD 



-46- 




FIG. 22 MOHR CIRCLE IN 3 PLANES 




Ps = STATIC STATUS 
Pa = ACTIVE FAILURE 
Pp = PASSIVE FAILURE 



aj+a-3 



FIG. 23 ACTIVE AND PASSIVE FAILURE 



-47- 



0.20 r- 



cr 



O 

a: 

u_ 

CO 
O 

o 

CO 

> 



y 0.05 

I 

ID 

o 

LU 




.4 6 

YIELD RATIO Xm/Xy 





■*-x 



k m 



8 



FIG.24 EQUIVALENT VISCOUS FRICTION VS YIELD RATIO 



-48- 



IV NON-LINEAR STIFFNESSES OF MODEL ELEMENTS 

During periods of low amplitude oscillation, a bridge-soil system 
can be modelled using the linear elements as described in Chapter II. 
It may be necessary in this case to treat the friction and expansion 
joint elements in a piece-wise linear fashion. For a severe earthquake 
however inelastic deformations may occur in the concrete columns and/or 
backfill soils, separations and impacts may develop between the abutments 
and backfills, slippages may take place in the expansion joints, and 
yielding may take place at the soil boundaries or in the foundation to 
complicate the behavior. 

It is the purpose of this chapter to describe the non-linear be- 
haviors of all elements and to derive their non-linear stiffnesses. 

A. GENERAL ELASTIC -PERFECTLY PLASTIC STRESS-STRAIN RELATIONS 

Presently, extensive literature . exists on the theory of plasticity 
and its application to different types of materials and structural ele- 
ments [30, 46, 85], Recently, its application has been extended to soil 
structures and frame structures as reported by Dibaj and Penzien, and 
by Porter and Powell 128, 83]. 

The first step in deriving the stress-strain relations for an elas- 
tic-perfectly plastic material is to assume a yield function expressed 
in terms of the stress space. This stress function can be expressed as 



f CT. .) = (49) 

ID 



-49- 



By application of the flow rule, the plastic strain increment tensor is 
derivable from this function using the relation 

6 e ij = x 8T77 (50) 

3-D 
where X is a non-negative scale factor. The total strain increment ten- 
sor is thus decomposed into its elastic and plastic components as ex- 
pressed by 

Se ( . = 6 e?. + 6 e*. (51) 

ID ID ID 

The generalized Hooke's law, relating the increment of stress tensor to 
the increment of elastic strain tensor, can be written in the form 

St.. = C^„ . 6 e E . (52) 

Using Eq. (51) , Eq. (52) can be rewritten as 

For an elastic-perfectly plastic material 

6 f = || — 6 T . . = (54} 

oT . . IT 

Substituting Eqs. (53) and (50) into Eq. (54) , one obtains 
3f -E 



3t. . C ijk£ 
ID 



« S* ■- « St' ■ ° l55) 



or 



^z — C... 6 e . - w- — C... kt X = (56) 

dT. . lnkJo k% dT . . i-jkx dT, „ 

lj J ID k ^ 



-50- 



Solving for A gives 



x - hc i jk ji !fr: 6£ k* < 57 > 

ID 



where 



1_ E 3f 9f 

h ■ c «u 3t.. Tr£ < 58 > 

Substituting Eq. (57) into Eq. (50) results in the relation 

6£ ij = hC mnkJl 8F- 6e kil 8777 (59) 

mn xj 

A further substitution of this relation into Eq. (53) gives 

St.. = c E <6e v0 -hC E . I| — 1| — 6 e ) (60) 

13 i]kJ6 kJl xjmn 9t.. 9t - mn 

or 

6 x.. = c E ., (6 e v0 - a. 6 e ) (61) 

lj xjkx, kx, k&mn mn 

where 

A, „ = hC.„ „ 7; r (62) 

kJlmn ijkJl 3t . . 3t 

J x] mn 

The final form of the stress-strain relation for elas to -perfectly plas- 
tic material can now be written in the form 



4 T i 3 " c ijk * 6 £ kji (63 > 



where 



C = C — C A (64) 

ijk£ ijk£ ijmn kilmn 



-51- 



and where 



P E 

C.. l0 = C.. A, (65) 

ljkA ljmn kicmn 



B. SOIL FINITE ELEMENT 

* 

•The Mohr-Coulomb criterion as stated in Eq. (43) of Chapter m 
may be expressed in terms of the principal stresses, namely 



(a - ) + (0 + a ) sin <{> = 2c cos A (66) 

13 13 

where > a > with tension as positive. In the case of two 
1—2—3 * 

dimensional stress in the x,y plane, this relationship takes the form 



{O + a ) sin <{> + 2R - 2c cos A = (67) 

x y 



where 



R = I ( X " Y ) 2 + • ( T xy )2] ^ (68> 



is the radius of the failure stress circle as shown in Fig. 25. 

1. Tangent Stiffness In Plain Strain - The derivative of tangent 
stiffness given by Eqs. (49) through (65) will now be presented in 
matrix form for the case of plain strain. Using the Mohr-Coulomb cri- 
terion as the yield function f(T. .), and the index notation for stresses 

ID 

such that T = a , T = a , and T = T , one obtains 
11 x 22 y 12 xy 

T - t 2 i. 

f = (T + T ) sin <f) + 2 [ (-^ — — ) + T 2 ]' 2 - 2c cos 6 = 

11 22 2 12 

(69) 



-52- 



and 



q 



3-D 



3f_ 

8t 



(70) 



ID 



Equation (50) becomes 



{Se P } 



X{q} 



(71) 



where 



{q>" 



< q > = < q q q > 
^ ^11 22 12 



T - T 
< [ — sin <p + ( - ) ] , [ — sin <p 



x - t 



[2-11] > 



(72) 



Equation (51) in matrix form can be written as 



• {6 e} = {8 e E + 6 e P } 



(73) 



where 



{6 e} = <6 z> = < 6 e 6 e 6 e > 

11 22 12 

{6 e P } T = < 6 e p > = < 6 e p 6 e P 6 e p > 



11 22 



12 



{6 e E } T = < 6 £ E > = < 6 e K 6 e E 6 e E > 

11 22 12 

{6 x} = [C E ] {<5 e E } 



and where 



{6t}=<6t> = <6t St 6t > 

11 22 12 



(74) 



[c E ] 



E 



(1+V) (1-2V) 



i-v 


V 





V 


l-V 











1-2V 




2 



(75) 



-53- 



A = h{q} T [c E ] {6 e} 



(76) 



where 



1 

h 


= £,*} 


[C E ] {q} 




E 
1+V 


sin <J) 
2(l-2v) 


1 
h = 


E B 
1+V 


<• 



2 2 R 2 



m sin 2 » 1_ + 3_ V 
2(l-2v) 2 2 D 2 

R 



(77) 

# 

(78) 
(79) 



Equations (62) through (65) in matrix form become 



[A] = h {q} {q} T [C E ] 



{6 T> = [C] {6 e} 



(80) 
(81) 



[C] = [C E ] - [C E ] [A] 

= [C E ] - h[C E ] {q} {q T } [C E ] 

= tc E ] - [c p ] 



(82) 



Letting 



{Q> 



{q> T [C E ] 



< > 

1+V *1 *2 *3 

E sin $ 

1+V * 2(1-2V) 



T " T -a 
11 12 sin 9 



T - T T 

11 22 12 



4R ' 2(1-2V) 



4R 



R 



(83) 



, the matrix [C ] can be expressed in the form 



-54- 



[C P ] = h[C E ] {q> {q} T [C E ] 



= h{Q} {Q}' 



B(l+V) 



Q Q 

2 1 

Q Q 

3 1 



Q Q 



C Q 

3 2 



Q Q 

13 
2 3 



(84) 



Finally, the tangent stiffness in explicit matrix notation becomes 



IC] = 



(1+V) (1-2V) 



1-V V 
V 




1-V 
1-V 



b(1+v) 



! A V,] 



Q Q Q ' 

2 1 2 



2 3 



3 1 3 2 3 



(85) 



2. Postulate for Application of Tangent Stiffness to the Case of 
Plane Stress - In order to use the previously derived tangent stiffness 
for the case of plane stress , one must make two assumptions as follows: 
(1) no yielding occurs in the third direction and (2) the results of 
triaxial tests are directly applicable to the case of plain stress. 

Under these assumptions, the above derivations for plane strain also 

E 
apply to the case of plane stress except that the matrix [C ] is changed 

to the form 



[C E ] = 



1-V' 



1 

V 




V 
1 
1-V 



(86) 



This results in the following changes 



-55- 



T - T 2 T 2 

{q} T [C E ] {q} = -22j [ (1+V) sin 2 c}> + (1-V) ( 1 - 4R 22 ) + (1-V) {-£) ] 



1 2EB 



h 1-V 2 



(87) 
(88) 



,-.. T - T 2 . T 2 

B = [ (1+V) sin 2 * + (1-V) ( " 4R 22 ) + (1-V) (-^-) ] (89) 



{Q> T = {q> T tC E ] 



— — < Q Q Q > 
l-v 2 2 2 3 

1 T - T 

< ~ (1+V) sin <f> + (1+V) 



1-V 



2 2 



4R 



T - T T 

j (1+V) Sin <J> - (1+V) " 4R " , (1-V) -12. > 



(90) 



[c p ] 



= h[C E ] {q} {q T } [C E ] 



= h{Q} {Q} 



T 



2B(l-v 2 ) 



Q Q Q Q 

12 13 



QQ Q ' 

2 1 2 



2 3 



QQ, QQ Q ' 

3 1 3 2 3 



(91) 



Finally, one obtains 



[C] = 



[c E ] 



tc p ] 



1-V 



1 V 

V 





1 
1-V 



2B (1+V) 



12 3 



2 1 2 



2 3 



QQ^ QQ Q ' 

3 1 3 2 3 



(92) 



-56- 



C. COLUMN ELEMENT 

1. Trilinear Yield Surface of the Moment-Axial Force Interaction 
Diagram - Moment-axial force interaction curves for the most commonly 
used sections of reinforced concrete columns are available in handbooks 
published by the American Concrete Institute [2] . These sections include 
the spirally reinforced, circular and square columns, and the symmetri- 
cally reinforced, rectangular tied columns. Three typical sections are 
shown in Fig. 27. For otner types of sections, a computer program has 
been developed to obtain the interaction curves by a direct analysis 
method [105] . 

There are three controlling points on the interaction curve which 
can be used to approximate its form using a trilinear relationship. 
These points are the minimum eccentricity point B, the balanced point C, 
and the pure moment point D, as shown in Fig. 26. Segment AB having a 
horizontal slope defines the ultimate axial load capacity as that axial 
load given by the ACI code for the minimum eccentricity condition [35] . 
Segment BC defining the compression failure zone connects point B with 
the balanced point C which corresponding to a concrete strain of 0.003 
and a steel strain equal to the yield strain. Finally, segment CD 
connects point C with point D which corresponds to the yield moment in 
the presence of no axial load. This latter segment defines the tension 
failure zone. Since tension seldom occurs in the columns of short 
bridges, it is not necessary to define the interaction diagram in the 
negative (tension) region of P. Line segment OD therefore is considered 
the boundary line for the yield surface which signifies zero tension 
capacity. 

-57- 



Using the approximate trilinear form of the interaction curve, the 
normalized yield stress function can be written as 



f. (S , S ) = a. S + b. S + c. = 
112 11 12 i 



(93) 



where 



s -2- 

1 p 2 



s =^L 

2 ' M 2 



(94) 



For line segments AB, BC, and CD, coefficients a., b., and c. are, 
respectively , 



M 



M 



a 
l 




1 




a 

2 


2 

P 


a 

3 


" M ~ 1 
2 

P 


b 
l 


— 





» 


b 

2 


2 


b 

3 


-1-5*- 

2 


c 

l 


= 


P 
l 

~ P 

2 




C 
2 


P, M 

1 1 

= P " M 

2 2 


C 
3 


P M 
_ _3 3 

P " M 

2 2 



(95) 



2. Tangent Stiffness - Using the trilinear interaction relation- 
ship, the derivation of the elastic-perfectly plastic tangent stiff- 
ness of a column follows the same procedure used previously for soils 
except one must consider that (1) plastic deformations are concentrated 
at the ends of the element with the deformations taking place indepen- 
dently over zero lengths at each end of the element, (2) plastic defor- 
mations are independent of the shear forces present, and (3) the stiff- 
nesses are expressed in terms of element end forces and displacements 
rather than in terms of stress and strain as in the case of soil ele- 
ments. In this case, one obtains 



-58- 



(du P > = 



du! 



du. 



{q} 1 {0} 



{0} {q} 



= tq] U> 



(96) 



P P 
where du and du are the plastic deformation increments at ends I and 
I J 

J of the element, respectively, and X and X are the associated pro- 
portionality factors. It follows therefore that 



tdu j = < du du du du du du > 
l 2 3 if 5 6 



W 



,3f: , 



m 



=<a. Ob. > ^ m = I or J 
l l m 



(97) 
(98) 



m 



{du} = 



{dS> = 



du. 



/du. 



dS. 



dS. 



,du~ 



'du. 



[K E ] 




(99) 



(100) 



where [K ] is the elastic stiffness matrix appearing in Eq. 23. Further, 
one obtains 



{X} = 



= [h] [q] T [K E ] {du} 



(101) 



[h]' 1 = [q] T [K E ] [q] 



[A] = [q] [h] [q] T [K E ] 



(102) 
(103) 



{dS} = [K] [du] 



(104) 



-59- 



[K] = [K E ] - [K E ] [A] 

= [K E ] - [K E ] [q] [h] [q] T [K E ] 

= [K E ] - [K P ] (105) 



3 . An Approximation Used in Numerical Iteration - Due to the 
occurence of impact upon the f rictional element at the interface of soil 
and abutment wall elements, the time interval used in a dynamic analysis 
to obtain a stable numerical solution must be quite small. It must be 
sufficiently small so that "overshooting" of the interaction diagram 
during a single interval as shown in Fig. 28 is minimized. Even though 
this interval is kept small, the error introduced into a solution by 
"overshooting* has been corrected [16, 60, 105] . 

In the present investigation, a simple procedure ' has been adopted 
for the overshooting. If, as shown in Fig. 28, the elastic stress 
state assumed during an interval moves the applied forces from point A 
to point B then a transition from an elastic to a yield state is indi- 
cated. While the new force vector S . as represented by point B is 
adopted without correction, point C which is the intersection of the 

new stress vector S . and the yield segment EF is used to calculate 

P 
the slope of the plastic deformation vector du . This same procedure 

is used when overshooting occurs at a discontinuity point on the inter- 
action curve as shown in Fig. 29. In this case, the elastic stress 
state assumed during an interval moves the applied forces from point A' 
to point B'. The new force vector S . intersects line EF; thus, the 
slope of EF is used to calculate the plastic deformation increment. 
Other investigators have used somewhat different procedures for this 



-60- 



correction [73, 77, 83, 107]. 



D. FRICTIONAL ELEMENT 



The non-linear behavior of the frictional element is described in 

terms of normal and shear stiffnesses k and k during three distinct 

n s 3 

stages, namely, (1) when reparation occurs, in which case k = k =0, 

n s 

(2) when compression occurs at the interface but the shear strength of 
the element is not exceeded in which case k and k are assigned high 
values, and (3) when compression occurs but the shear strength of the 

element is exceeded in which case k = and k retains a high value. 

s n ' 

The shear yield strength of the element can be defined by the Mohr- 
Coulomb yield criterion, i.e., 



T = a tan <J> (106) 

T w 

where <J> is the angle of wall friction between the soil and the abut- 
ment wall. 

Before discussing the value of <f> to be used, two terms must be 

Vr 

defined [63] . Firstly, the constant volume frictional angle is defined 
a - O 

— 1 1 3 

by d) = sin ( ) , where O and O are the axial stress and 

cv a + O cv 1 3 

1 3 

confining pressure, respectively, in a triaxial test at that stage when 
the sand strains without further volume change. Secondly, peak friction 
angle <J> is defined as the slope of Mohr envelope which is a function of 
the stresses indicated at the peak of stress-strain curve of a triaxial 
test. 

For backfill against a concrete wall, Lamb suggests that the angle 
of wall friction <b is about equal to <J> and that it typically has a 

-61- 



numerical value of about 30° [63] . Seed and Whitman in a discussion of 
dynamically active pressure against walls suggest that ({) = <|)/2 is 
satisfactory for most practical purposes [97] . 



E. EXPANSION JOINT ELEMENT 



The stiffnesses of the expansion joint element can be defined in 

terms of normal and shear stiffnesses k and k during three stages, 

(1) when the frictional resistance between the deck and abutment is not 

exceeded in which case k and k are assigned high values, (2) when the 

n s 

frictional resistance is exceeded, but the gap between deck and abut- 
ment as shown in Fig. 6a is not closed, in which case k = and k 

s n 

retains a high value, (3) when the frictional resistance is exceeded 
and the gap is closed in which case, if the relative displacement indi- 
cated is consistent with gap closure, high values k and k are retained, 



F. SOIL BOUNDARY ELEMENT AND EQUIVALENT COLUMN OF FOUNDATION 

The non-linear behavior of the soil boundary element and the equi- 
valent column foundation element can be approximated by defining yield 
levels using standard methods and adopting elasto-plastic hysteretic 
models. In the present investigation, only elastic behavior has been 
considered. 



-62- 



Y R VT 


2 1 x * / 


1 




^"^"^ C 


\ 


•^ : ^ 


-" t 



1/2 (cx x +o- y ) 
FIG, 25 MOHR -COULOMB YIELD FUNCTION 




D (0,M 3 ) 



^> M 



FIG. 26 COLUMN INTERACTION CURVE 



-63- 




(a) CIRCULAR SECTION WITH BARS CIRCULARLY ARRANGED 




8-#IO 



(b) SQUARE SECTIONS WITH BAR CIRCULARLY ARRANGED 




*»M 



tc) RECTANGULAR SECTION 
FIG. 27 INTERACTION DIAGRAM OF CONCRETE COLUMN 



-64- 




*-M 



FIG. 28 SITUATION AT OVERSHOOTING 




*-M 



FIG. 29 OVERSHOOTING AT DISCONTINUITY 



-65- 



V DYNAMIC ANALYSIS PROCEDURES 

In the subsequent sections, the nonlinear coupled equations of mo- 
tion for the discrete parameter soil-structure system are formulated 
and the method used to establish their associated stiffness, mass, and 
damping matrices are described. Also presented are the step-by-step 
integration techniques employed in their solution. 

Ac EQUATIONS OF MOTION 

Although some previous investigations have considered spacial vari- 
ations in the earthquake ground motions [27, 32, 43] , the present inves- 
tigation assumes identical motions at all base points of the soil-struc- 
tural system. This assumption is considered reasonable due to the rela- 
tively short lengths of bridges being considered. 

The coupled equations of motion at time t for an N degree of free- 
dom system subjected to rigid base excitation can be expressed in the 
matrix form 

[M] {u> + [CI {u> ■+ [Kl {u> = {r> (107) 

t t t t t 

where [M] is the constant mass matrix, [C] , and [K] are the time 
dependent damping and stiffness matrices, respectively, and where {u} , 
{u} , and iu} are the nodal point displacement, velocity, and accelera- 
tion vectors, respectively. The excitation force vector "Cr). due to 
rigid base motions is given by the relation 

{r> = - [M] {i} ( u* + u Y _ ) (108) 

t gt gt 

-66- 



where {i} is the unit vector and u ^ and u ^ are the horizontal and ver- 

gt gt 

tical components of ground acceleration. 



Bo STIFFNESS MATRIX 

The complete stiffness matrix [K] is assembled from the individual 
element stiffness matrices using the direct stiffness method [22] . The 
individual element stiffnesses during the elastic and inelastic ranges 
in each time interval are obtained by the procedures described in Chap- 
ters II and IV. The complete stiffness matrix takes on a symmetric 
banded form; thus, only the diagonal and the off -diagonal terms on one 
side need be stored in the computer. 

Co MASS MATRIX 

The mathematical model used assumes all mass as concentrated at 
the nodal points. The diagonal mass matrix which results represents a 
significant saving in computer storage and computational time when com- 
pared with similar requirements for the consistent mass matrix 121] . 
One-third of the mass of each triangular element, one-fourth of the mass 
of each quadrilateral element, and one-half of the mass of each beam 
element are lumped at their respective nodal points. No rotational 
moments of inertia are assigned to these masses. The resulting mass 
matrix for the complete soil-structural system takes the form 



IM] = diag < M M M > (109) 

12 n 

where M. is the mass associated with the ith degree of freedom and n 



-67- 



is the total number of degrees of freedom present in the system. The 
static condensation procedure is used to eliminate the degree of free- 
dom of zero rotational masses in the solution of eigenvalues. 

D. DAMPING MATRIX 

Various methods have been used by investigators to determine the 
viscous damping matrix corresponding to matrix IC] in Eq. (107) 144]. 
Wilson and Penzien have described two methods for evaluting orthogonal 
damping matrices [115] . The first method relates the modal damping 
ratios to the coefficients in the Caughey series form [15] . The second 
method is a direct approach which expresses the damping matrix as a sum 
of a series of matrices each of which produces damping in only one par- 
ticular mode. The second approach has the advantage that prescribed 
damping ratios in all modes are easily controlled. 

The Rayleigh damping matrix which constructs a damping matrix from 
a scaled linear combination of the mass and stiffness matrices is used 
in the present investigation. This type of damping matrix has the ad- 
vantage that it can be calculated directly using the relation 

[CJ t - cc[M] + 3[K] t (110) 

where a and 8 are scalar quantities to be prescribed. By properly 
selecting these scalar values, the damping ratios can be controlled in 
two normal modes. It can be shown that these quantities are related to 
the damping ratios (£) and circular frequencies (u)) of the ith and jth 
normal modes through the equations 



-68- 



2wio (£ .to. - £ to ) 
ffl = i j ^j i ^ij' (111) 

(a). 2 - a). 2 ) 
i D 

2 (£.0). - £.w.) 

3 = i-J: 3-J- 

(oo. 2 - to. 2 ) 



(112) 



Further, it can be shown that if a and 3 satisfy Eqs. (Ill) and (112), 
the damping ratio in nth normal has the value given by 

<* + 3 to 2 . ..... 

c- _ n_ (113) 

^n 2 (0 
n 

In the present investigation, the numerical values of a and 3 are deter- 
mined by using the initial elastic soil-structural system and by pre- 
scribing the damping ratios of any two modes of the system. These quan- 
tities are then held constant at these values throughout the time 
history of response including those periods of time when the system 
responds inelastically. 

As shown by Eq. (107) , the stiffness matrix varies with time due 
to nonlinear effects; therefore, the damping matrix also varies with 
time. Because the stiffnesses in the system decrease considerably 
during periods of element yielding, the viscous damping present during 
these periods also decreases. It should be kept in mind, however, that 
the major sources of energy dissipation during these periods are the 
hysteresis loops in the force-deformation relations as described in 
Chapter III. 

E. STEP-BY -STEP INTEGRATION TECHNIQUES 



-69- 



Having the solution of the coupled equations of motion at time 
t, the step-by-step integration procedure allows one to obtain their 
solution at a later time t+At. To' develop this procedure, the matrix 
equation of motion is transformed to its incremental form by subtract- 
ing Eq. (107) for time t from the corresponding equation for time t+At 
as given by 

[M] ({u}. + {Au}.) + [CI ({u> + {Au> ) + [Kl ({u> + {Au}j 
t t tt t tt t 

= {R> t + {AR> t (114) 

In this equation, the incremental quantities represent those changes 
taking place during the interval At following time t. Thus, one ob- 
tains the incremental form 

[M] {Auh + [CI {Au> + [Kl {Au> = {Ar> (115) 

t t t t t t 

To find the incremental changes, various procedures can be employed 
[7, 76] . The differences in these procedures relate to the analytical 
form of the variation in response over the time interval At. In the 
present investigation, two different analytical forms have been pro- 
grammed for computer solution, namely, constant acceleration and linear 
acceleration. These forms lead to the following equations for velocity 
and displacement at time t+At expressed in terms of the state vectors 
at time t and the acceleration vector at time t+At: 

Constant Acceleration 

{i} t + At " {i} t + J 4t m t + \ At K} t + At (116) 



-70- 



{u) t + at - {u} t + &t {i} t + \ At2{u> t + ? At2{a t + at (117 » 



Linear Acceleration 

«W " {i} t + J &t {U >t + 5" At ' K >t + At tll8) 

< U W - {u} t + At to t + 1- ^ ta t - 1- At 2 «} t+it (119) 

Using the following definitions for the incremental vectors 



Uu} t - {u} t+it - K} t (120, 

{Au} t = {i} t+At - «} t (121) 

{Au} t = {u) t+it - {u} t ' (122) 



the incremental velocity and acceleration vectors can be expressed 
in the form 

Constant Acceleration 

{All) = — {Au> - {a}. (123) 

t . At 2 t t 

(Au> t = ^| {Au> t - (B} t (124) 



where 



<Ah = ~ {u> + 2 {u> (125) 

t At t t 

{B> t = 2 {ti> t (126) 



-71- 



Linear Acceleration 

{Au> = — {Au> - { A > (127) 

fc At 2 t t 

' {A ^ } t = At {Au} t " {B} t (128) 

where 

M t - jf «> t ♦ 3 {5} t (129) 

{B> t = 3 {u) t + j At {u) t (130) 

Using these relations, the incremental equation of motion, Eq. (115) , 
can be written in the form 



[K] {Au} = {AR} (131) 



where 



{Ar} = {AR> + [M] {a> + C ■ [M] {b> (132) 

t t t 2 t 



[Kl = >[Kl + C [M] (133) 

t t i 



and the actual incremental displacement vector can be expressed as 



{Au> = C ({Au> + 3{B> ) (134) 

t 3 t t 



Constants C , C , and C are given by the relations 



Constant Acceleration 



C = 2 a At + 4 (135) 

1 At 2 + 2 3 At 

c = a - c B (136) 

2 1 



-72- 






c , - at4V < i37 > 



Linear Acceleration 

C - 3 a At + 6 (138) 

1 At 2 + 3 3 At 

c = a - c 3 (139) 

2 1 

At 

(140) 



3 At + 33 ■ 

After computing the incremental displacement vector using Eq. (134) , 
the corresponding incremental acceleration and velocity vectors are 
determined using Eqs. (123) and (124), or Eqs. (127) and (128), respec- 
tively. The displacement, velocity and acceleration vectors at time 
t+At are then evaluated using Eqs. (120), (121) and (122), respectively. 
The tangent stiffness, strain, and stress for each element can now be 
calculated for time t+At. 

An alternative solution of the incremental equilibrium equation, 

Eq. (115) can be obtained by separating the tangent stiffness matrix 

E P 
IK] into its elastic and plastic parts, [K ] and [K ] , as described 

in Chapter IV. That term associated with plastic deformation is then 

transferred to the right hand side of the equation of motion and is 

treated as an equivalent load vector [21] . 

In the above described step-by-step integration procedures, the 

initial displacements and velocities at time t = are assumed equal 

to zero. The very first incremental acceleration vector is then 

computed directly from Eq. (115), i.e., 

• [M] lAu>+ [C]„{0}+ [K] (0} = {AR> 
u o u o 



-73- 



or 

{Au} = [M]" 1 {AR} = [M]" 1 {R} = -{u } 
o o o go 

The initial forces existing in the elements cannot be assumed zero but 
must be taken equal to the static gravitational forces since tangent 
stiffness is dependent upon total force (gravity plus seismic) . 

While step-by-step integration procedures similar to those described 
above must be used for nonlinear analyses/ they may or may not be used 
for linear analyses since the mode superposition method is an alternate 
method which can be used for linear analyses [116] . In the present in- 
vestigation, it was found computationally convenient to use the step- 
by-step method for both linear and nonlinear analyses. 

F. TIME INTERVAL At 

The step-by-step integration method is accurate only if the time 
interval At is small compared with the shortest period T of the soil- 
structural system and is also small compared with the predominant pe- 
riods in the excitation. Assuming the latter condition is satisfied, 
the ratio At/T must be selected less than a certain critical value to 
insure a convergent and stable solution in the case of the linear acce- 
leration method; however, it can be shown that the constant acceleration 
method is always stable for a linear system [76] . In the present study, 
the presence of the nonlinear friction elements tend to encourage an un- 
stable response, if the At/T ratio is taken too large. Therefore, ex- 
treme care must be taken in selecting the numerical value of this ratio. 
The effects of this particular parameter on dynamic response are discussed 



-74- 



subsequently in Chapter VI. 

G. EARTHQUAKE INPUT 

In the present investigation, the horizontal ground motion was 
prescribed in accordance with the acceleration time-history shown in 
Fig. 30. This artificial accelerogram was generated by A. K. Chopra to 
simulate the ground motions produced by the San Fernando earthquake at 
the site of the Olive View Hospital located about 6 miles southwest of 
the epicenter 118] . It has a peak acceleration of 0.5 g and a uniform 
phase of high intensity shaking for 8' seconds. 

The vertical ground motions were assumed zero for the present study , 
but the computer program has the option to permit input of vertical 
ground motions. 



-75- 



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UJ 




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1- 


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0l (9) 130DV 



-76- 



VI PARAMETER STUDIES 

The previously defined mathematical modelling and dynamic analysis 
procedures have been applied to a straight version of an existing slightly- 
curved skewed bridge, namely, the North Connector Undercrossing located 
approximately 800 ft. northerly of Route 5 - San Fernando Road Interchange 
in the city and county of Los Angeles. Plan and elvation views of this 
bridge are shown in Fig. 31 along with cross-sectional views of the decks 
and the centrally located supporting columns. Figure 32 shows a sectional 
view of the left abutment with a portion of the backfill of 'extent L and 
height H. 

It is the purpose of this chapter to present the results of dynamic 
analyses for the North Connector Undercrossing when subjected to one com- 
ponent of earthquake excitation in its longitudinal direction. The ground 
motions used in this study were the Olive View Hospital accelerations 
shown in Fig. 30 during the time interval 2-4 seconds. This relatively 
short duration was chosen to minimize computer time and yet to provide 
sufficient time for a representative dynamic response to occur. The 
peak acceleration in this excitation is approximately 0.3 g. 

In order to establish an appropriate mathematical model of the en- 
tire bridge-soil system, parameter studies were first carried out using 
a rigid wall with uniform elastic backfill. The complete bridge-soil 
system was then analyzed with certain parameter variations. The results 
of these studies along with the results of an integration-time-interval 
sensitivity analysis are presented in the subsequent sections of this 
chapter . 

-77- 



A. RIGID WALL-BACKFILL SYSTEM 

1. Lateral Extent of Backfills - To study the longitudinal dimen- 
sion or extent of backfill required in the mathematical modelling of 
bridge-soil systems, analyses were conducted using a single rigid wall 
and a uniform linear -elastic backfill having the properties y - 110 pcf , 
V = 0.35, and G = 10 ksi. Five cases with L/H ratios equal to 4.8, 6.0, 
10.0, 14.0, and 22.0, as shown in Fig. 33, were used in this investigation, 
The finite element idealization consists of 5 elements in the vertical 
direction and a variable number in the horizontal direction. The ele- 
ment length to height ratio R = a/b was maintained at a constant value 
equal to 2 throughout the extent of the backfill in Cases 1 and 2 but 
was maintained only over a distance 2H from the wall in Cases 3, 4, and 
5. This ratio was set equal to 4 for all elements beyond the distance 
2H in the latter cases. No friction elements were placed between the 
rigid wall and the backfill in these particular studies and the left 
vertical boundary of the backfill was assumed free in each case. The 
fundamental period of the soil system under these conditions is about 
0.08 seconds in each case with slight increases occurring with increas- 
ing values of L/H as shown in Table 1. 

Assuming rigid body earthquake excitations to occur along the en- 
tire base of the backfill and at the rigid-wall vertical-boundary and 
assuming 5 percent of critical damping in the first two modes of vibra- 
tion, time histories of response were obtained for all 5 cases. Response 
quantities of greatest interest were the maximum or peak values of (1) 
total lateral dynamic force exerted on the rigid wall, and (2) horizontal 
displacements and accelerations at various location in the backfill. 

-78- 



Figure 34 shows the maximum total lateral dynamic force F, exerted 

d 

on the wall for each of the 5 cases studied. This force is reasonably 

constant at a value of approximately 14.5 kips over the range studied; 

i.e. 4.8 < L/H < 22, and is about 3.9 times greater than the average 

static value which is approximately 3.7 kips. The mean location of the 

resultant force F, was found to be at a distance 0.52 H above the base 
a 

which is considerably higher than the position of the resultant static 

force F located at 0.34 H. 
s 

The maximum horizontal displacements relative to the moving base 
at 6 different locations in the backfill are shown in Fig. 35 for all 
5 cases. While these displacements are reasonably constant over the 
range of L/H studied, large differences are noted from one point to 
another. The displacements for points 1 and 2 are very small due to the 
fact they are located only one element away from the rigid wall. Points 

3 and 4 which are 5 elements away from the wall experienced much larger 
displacements than points 1 and 2 and points 5 and 6 at the left bounda- 
ry experienced even larger displacements. These relative displacements 
reflect the manner in which the rigid wall boundary effects decay with 
increasing distance from the wall. The displacements for points 3 and 5 
are considerably larger than the corresponding displacements for points 

4 and 6, respectively; thus, demonstrating the increase in displacements 
with vertical distance above the rigid base. 

Figure 36 shows the maximum total horizontal acceleration at loca- 
tions 1-6 for all 5 cases studied. The relative variations of this 
response quantity with L/H, horizontal distance from the wall, and ver- 
tical distance above the base are similar to those previously described 



-79- 



for horizontal displacement. It should be noted that the average accele- 
ration with L/H ranges from about 0.5 g for points 1 and 2 to about 1.1 g 
for point 5. These values represent a rather small amplification of 
acceleration at points 1 and 2 and a fairly large amplification of the 
acceleration at point 5 over the peak excitation acceleration of about 
0.3 g. 

Based on the numerical results shown in Figs. 34-36 and summarized 
in Table 2, it appears that an L/H ratio equal to 14 is sufficient for 
use in any bridge-soil analysis. This ratio may even be reduced to a 
value of 6 with little loss in accuracy in determining the maximum values 
of abutment backfill forces. However, this reduction could introduce 
significant changes in the predicted maximum horizontal accelerations. 

2. Length to Height Ratio of Finite Elements - Length to height 
ratios R = a/b for the finite elements of the backfill model were rather 
arbitrarily assigned values as shown in Table 1 for the 5 cases pre- 
viously defined. To investigate the influence of changing these ratios, 
4 new cases as shown in Fig. 37, each having an L/H ratio equal to 10, 
are defined. The length to height ratio equals 2 throughout the model 
for Case 1 and over a distance 2H from the wall for Cases 2, 3, and 4. 
Beyond 2H, the length to height ratio equals 4, 6, and 10 for Cases 2, 
3, and 4, respectively. Five elements are again used in the vertical 
direction of the model and fixed and free boundaries are prescribed at 
the right and left ends, respectively. Linear elastic soil properties 
are again assumed equal to the values previously assigned. 

As before, rigid body earthquake excitations are assumed to occur 
along the entire base of the backfill and 5 percent of critical damping 

-80- 



is assigned to the first two modes of vibration. Time histories of 
response are obtained for all 4 cases, including (1) total dynamic force 
exerted on the rigid wall, and (2) horizontal displacements and accelera- 
tions at 6 locations in the backfill. 

Figure 38 shows total static force F and the maximum total dynamic 

lateral force F, exerted on the wall for each of the 4 cases defined 
a 

above. This force is fairly constant at a value of approximately 14.5 
kips over the range of R assigned beyond x = 2H and is about 3.9 times 
greater than the average static* value. The mean location of the resul- 
tant force F, is at a distance about 0.52H above the base. The average 
a 

position of the static resultant is 0.34H. 

The maximum horizontal displacements relative to the moving base 
and' the maximum total horizontal accelerations at the 6 different loca- 
tions are shown in Figs. 39 and 40, respectively, for all 4 cases. The 
results in these figures are quite similar to the results shown the 
corresponding Figs. 35 and 36. Thus, it is apparent that the changes 
introduced in R for x > 2H have introduced relatively small changes 
in overall response and that Case 4, Fig. 37, can be considered a 
reasonable model of the backfill. 

Table 3 presents a summary of maximum response for the above des- 
cribed 4 cases. 

3. Number of Finite Elements in the Vertical Direction - All pre- 
vious cases analyzed have used 5 finite elements in the vertical direc- 
tion of the backfill. To check the adequacy of this number the distribu- 
tions and resultant magnitudes of the static and dynamic backfill pres- 
sures on the rigid retaining wall are compared using 5 and 10 elements 

-81- 



in this direction. 

Two cases as defined in Fig. 41 are used for this comparison. Both 
cases use an L/H ratio equal to 10 and R ratios equal to "2 and 10 for 
x < 2H and x > 2H, respectively. Case 1 uses 5 elements in the vertical 
direction while Case 2 uses 10 elements. 

t It is quite clear from the results in Fig. 41 that the distributions 
and maximum resultant magnitudes of the' backfill forces are very similar 
for Cases 1 and 2 and that the positions of the corresponding resultants 
almost coincide. Considering this fact and the fact that their time 
histories (see Fig. 42) are very similar, one may conclude that 5 finite 
elements in the vertical direction of the backfill is sufficient for 
engineering purposes. 

4. Soil Stiffness - To study the influence of soil stiffness on 
the dynamic response of the backfill soil system, the finite element 
model identified as Case 4 in Fig. 37 was analyzed for three soil con- 
ditions, namely, a uniform soil modulus equal 10 ksi, a uniform soil 
modulus equal to 2.5 ksi, and a variable soil modulus in accordance 
with Eq. (38) for K =50. Poisson's ratio V was assumed equal to 0.35 
and the unit weight was again assigned the value 110 pcf . These new 
soil conditions are identified as Cases 1 - 3 in Fig. 43. 

While the shapes of the distributions of maximum total wall pressures 

and their resultant force positions are quite similar for all three cases, 

the magnitudes of the resultant dynamic wall pressures vary considerably 

with the soil stiffness condition. This variation ranges from F,= 

(100) (13.06) for Case 3 to F = (154.0) (13.06) for Case 2 . Obviously, 

a 

for the particular excitation used, the less stiff backfill soils produce 

-82- 



higher backfill forces on the rigid wall. If an excitation having a 
different amplitude distribution with frequency had heen used, this ob- 
servation could be changed considerably. Therefore, one must use cau- 
tion in interpreting the results of this particular study. It is impor- 
tant to recognize however that soil stiffness can be an important para- 
meter which should be studied using realistic earthquake excitations. 

5. Frictional Element Stiffnesses - As pointed out previously, the 
shear and normal stiffnesses of the frictional elements located between 
backfill soil and abutment walls are arbitrarily assigned finite but 
very high values rather than infinite values to avoid discontinuities 
in their force displacement relations. To study the influences of these 
stiffnesses on dynamic response, the rigid wall-soil system shown in 
Fig. 44 was analyzed assuming linear soil behavior. In these studies 
the shear and normal stiffnesses (K and K ) for each frictional element 
were assigned equal values ranging from 1 to 10 ksi. 

The total static and total maximum dynamic lateral wall forces 
(F and F ) obtained in these studies are plotted in Fig. 45 for stiff- 
nesses K = K_ equal to 1, 10 , 10 b , and 10 ksi. As one would expect, 
these forces are reasonably constant over a wide range of stiffnesses. 

However, as the stiffnesses approach zero, F and F, also approach zero 

s d 

which represent unrealistic values. Theoretically both F and F, should 
approach constant values asymptotically with increasing stiffnesses, 
however the value of F for K = K = 10 ksi is much larger than for 
the smaller values of stiffness. This large increase is due to a numeri- 
cal instability which developed in the analysis procedures and therefore 



-83- 



should be ignored. It appears therefore that realistic wall forces can 
be obtained by selecting stiffnesses in the range 10 < K = K < 10 
ksi. 

Although the asymptotic static value of wall force in Fig. 45 is 
consistent with values previously presented for cases having no friction 
elements, the maximum dynamic force of about 40.0 kips is considerably 
larger than the average value (14.5) previously presented. This increase 
in dynamic wall force is due to the separations and associated impacts 
which occur between the backfill soil and the upper part of the rigid 
wall. Thus it appears that for high intensity excitations, the friction 
element is essential to realistic modelling. 

B. INTEGRATION-TIME-INTERVAL SENSITIVITY ANALYSIS 

Throughout the rigid wall-backfill parameter studies previously 
described, the numerical integration time step was assigned a value 
equal to 0.01 seconds and the constant acceleration method was used which 
enables response to be stable, but not necessarily convergent, for all 
modes of vibration. To study the adequacy of using 0.01 seconds for At, 
the rigid wall-soil system defined by Case 4, Fig. 37, was re-analyzed 
using At = 0.001 seconds. The total number of degrees of freedom for 
this system is 90 with the fundamental period being 0.084 seconds and 
the highest period estimated at 0.0065 seconds. The convergent limit 
of the ratio of time step duration to period, i.e. At/T, is 0.39 for the 
constant acceleration method. Thus, for At = 0.01 seconds, the conver- 
gent period T is 0.026 seconds. Since this period corresponds to the 
period of the 22nd mode of vibration, only the lowest 22 modes are 

-84- 



convergent in the constant acceleration method of analysis when At = 0,01 
seconds. If on the other hand, At = 0.001 seconds, all modes are con- 
vergent . 

To check the accuracy of response obtained for the above case using 
At = 0.01 seconds, the time histories of lateral dynamic wall force and 
horizontal acceleration at point No. 1 (see Fig. 40) are obtained for 
At = 0.001 and are plotted in Fig. 46 where they can be compared with 
corresponding results for At = 0.01 seconds. The vertical acceleration 
time histories for point No. 1 are plotted in Fig. 47 for At = 0.01 
and 0.001 seconds. A summary of the maximum values of response for 
this case is presented in Table 4. 

Obviously, the results of Fig. 46 indicate that a time step inter- 
val of 0.01 seconds is quite adequate in predicting total lateral wall 
force and horizontal acceleration time histories. However, the results 
of Fig. 47 indicate the very low level vertical acceleration time his- 
tories caused primarily by very high frequency modal responses cannot 
be predicted accurately by At = 0.01 seconds. Since this high frequency 
response is relatively unimportant from an engineering point of view, 
it is concluded that At = 0.01 seconds is adequate for the previously 
described rigid wall-soil parameter studies and also for the bridge-soil 
system studies to be described subsequently. 

C. BRIDGE-SOIL SYSTEM 

To study the dynamic response of the combined bridge-soil system, 
3 mathematical models of the North Connector Undercrossing as shown in 
Fig. 48 were defined. Model A has fixed boundary conditions at depth H 

-85- 



of the backfilis, at the base of abutments, and at the base of all col- 
umns. Model B has fixed boundary conditions at depths 2.2H and 2.5H of 
the backfills, leveling with bases of pier columns* which allows the base 
of abutments to translate and rotate with the soil system, and fixed 
boundary conditions are also provided at the base of all columns. Model 
C has fixed boundary conditions only along the base of the backfills as 
in Model B. The bases of abutments and columns of this model are attached 
to equivalent columns representing the foundation flexibility. These 
equivalent columns of course, have fixed boundary conditions at their 
bases. 

The soil elements in all three models can be assumed linear or non- 
linear as desired and friction elements can be included in Models A and 
C, but not in Model B. The backfills extend a distance 6H in all models 
as shown. The bridge deck is linear in each model; however, either 
linear or nonlinear columns can be used. Backfill and foundation soils 
were assumed to have the properties G = 10.0 ksi, V = 0.35, y = 110 
lb/ft 3 , c = 0, and <f> = 30°. 

1. Soil Pressures on Abutments - The static and maximum dynamic 

pressure distributions on one abutment wall for two cases are shown in 

Fig. 49. The model used for Case 1 is Model A with no friction elements 

and with all other elements assumed linear. The model used for Case 2 

is Model B in its complete linear form. Due to the characteristic 

response of the bridge-soil system, the static and dynamic pressure 

distributions are quite different in form in each case. The resultant 

lateral static force F for Case 2 is about 17% less than for Case 1, 

s 

due to the change in abutment flexibility and the maximum resultant 

-86- 



dynamic force F for Case 2 exceeds the value for Case 1 by 170%. This 
latter difference is undoubtedly due to a closer matching of the lower 
mode periods of vibration for Case 2 with the predominant periods in the 
excitation. The location of the resultant static force is at about H/3 
for Case 1 and at about H/2 for Case 2 while the location of the maxi- 
mum dynamic resultant force is at about 0.53H and 0.6H, respectively. 

Another check on the influence of bridge structure flexibility on 
the resultant abutment soil forces can be made by comparing the results 
for Cases 1 and 2 in Fig. 50. In this figure. Case 1 is identical to 
Case 1 in Fig. 49 but Case 2 is different. Here, Case 2 is actually the 
same as Case 1 except that the abutment wall and column stiffnesses 
have been reduced by a factor of 10. It is seen that Case 2 shows a 75% 
increase in the maximum dynamic resultant force over Case 1 due to the 
decrease in bridge structure flexibility. This increase is consistent 
with the similar increase previously noted for Case 2 in Fig. 49. The 
location of the resultant dynamic lateral force is again at about mid- 
height. The increase in the resultant static force for Case 2 over the 
value for Case 1 is due to the increase in rotation (due to deck dead 
loads) at the top of the abutment caused by the reduced abutment flex- 
ibility. 

Further results of analysis are shown in Fig. 51 identified as Cases 
1, 2, and 3. In this figure, Case 1 is again the complete linear ver- 
sion of Model A. Cases 2 and 3 are also based on Model A but Case 2 
has introduced one nonlinearity, namely the friction elements, and Case 
3 has employed two nonlinearities - friction elements and nonlinear 
soil elements. The very large increase in maximum dynamic force F for 



-87- 



Cases 2 and 3 over Case 1 is due primarily to the impact wall forces 
following separations between wall and backfill. 

Finally the results of two additional analyses are s,hown in Figure 
52. Cases 1 and 2 in this figure are based on Model C using friction 
elements and linear soil elements; however, Case 2 uses the equivalent 
foundation columns while Case 1 does not. The most significant result 
to note in Fig. 52 is that force F is more than twice as great for Case 
2 over the value shown for Case 1. Again this increase is due to the 
fact that the more flexible bridge system has a closer matching of fre- 
quencies with the predominant frequencies in the excitation. 

2. Total Seismic Force Carried by Columns and Abutments - To inves- 
tigate the maximum total base shear carried by columns and abutments, 
five cases were analyzed as indicated in Table 5. Case 1 represents the 
bridge alone with no soil-structure interaction, i.e. Model A, Fig. 48, 
but with no backfill; Case 2 is Model A with linear backfill and no fric- 
tion elements; Case 3 is Model .B* Fig. 48, with linear backfill; Case 4 
is Model A with linear backfill and friction elements; and Case 5 is 
Model A with non-linear backfill and friction elements. Clearly, the 
presence of backfill contributes significantly to the maximum total base 
shear, also it appears from Case 3 that the total base shear increases 
with overall flexibility which is again evidence of a better matching of 
the lower natural frequencies with the predominant frequencies in the 
excitation. The relatively large displacement shown for Case 3 is due to 
the large flexibility of the system for this case in comparison with the 
other cases. 



-88- 



Do COMPARISON OF RESULTANT ABUTMENT BACKFILL FORCE OBTAINED BY ANALYSIS 
AND THE MONONOBE-OKABE METHOD 



One commonly used formula in calculating the resultant dynamic lateral 
force on the abutment wall is the Mononobe-Okabe formula [53, 55, 97] . 
This formula has the following form 

P p = {1 - K v ) * r ' x * V (141) 

where 

P = Passive earthpressure at depth x 

K = vertical seismic coefficient 
v 

Y = Unit weight of soil 

x = arbitrary depth 

K_ = Passive earthpressure coefficient during earthquake 

The coefficient K is given by 

. cos 2 (<J>-e +8) 



Kg r /sin $ • sin(<f>+a-U ) "| 2 

cos(6-a) 



[/sin <p • sin 
1 - / , Q Q . — 
cos (6-0 ) • 



(142) 



where 



<J> = Angle of friction of soil 
— l ^h 

e o = tan I3T 

v 

1L = Horizontal seismic coefficient 

= Angle between the backline of the wall and the vertical 
line 



-89- 



a = Angle between ground surface and the horizontal line 

'o - tan IT^o 



Using <p = 30°; = 0°; 9 = tan * ,°' 1 n = 6°; a = 0°; y = 110 lb/ft 3 



K =2.86 

The earthpressure at bottom of wall is 

p = (1-0.0) * 110 * 13.5 * 2.86 = 4.25 K/ft 2 
P 

and the total lateral force on the wall is 
F J = i * 13.5 * 4.25 = 28.6 K/ft ' 

Using the non-linear form of Model A, Fig. 48, or Case 3 of Fig. 51 , 
analysis gives 

P. = 35.0 K/ft 

a 

■This analytical result is a higher value than that formula given by the 
Monobe-Okabe . The position of the resultant force is assumed a dis- 
tance H/3 above the base when using the formula; however, the analysis 
shows it to be 0.44H above the base. This higher position given by an 
analysis is consistent with other investigations [97, 117]. 



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-117- 



VII GENERAL CONCLUSION 

Based on the results of this investigation, it is concluded that 
soil- structure interaction effects must be considered when analyzing the 
dynamic response of short, stiff, single or multiple span bridges. The 
mathematical modelling and computer programs presented herein provide an 
effective means of conducting such analyses. 

Since the numerical results obtained in this investigation are very 
limited, caution should be exercised when interpreting them in a quanti- 
tative sense. Further analyses are recommended to complete the parameter 
studies. 



-118- 



BIBLIOGRAPHY 



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-120- 



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64. Lastrico, R. M. (1970) 

"Effects of Site and Propagation Path on Recorded Strong Earth- 
quake Motion," Ph.D. Thesis, University of California, Los Angeles. 

65. Lew, H. S., Legendecker, E. 7., and Dikkers, R. D. (1971) 
"Damages to Bridges and Highways," Chapter 7 of United States, 
Department of Commerce, Building Science Series 40, Dec. 1971 .. 

66. Lysmer, J,, and Kuhlemeyer, R. L. (1969) 

"Finite Dynamic Model for Infinite Media," Journal of the Engineer- 
ing Mechanics Division, ASCE, Vol. 95, No. EM4, proc. paper 6719, 
August 1969, pp 859-877. 

67. Lysmer, J., and Duncan, J..-M. (1969) 

"Stressea and Deflections in Foundations and Pavements," Department 
of Civil Engineering, Institute of Transportation and Traffic 
Engineering, University of California, Berkeley, 4th Ed. 1969. 



-124- 



68. Matsuo, H. (1941) 

"Experimental Studies on the Distribution of Lateral Earthpressure 
in Earthquake," Journal of the Japan Society of Civil Engineers, 
Vol. 27, No. 2, 1941. 

69. Matsuo, H., and Ohara, S. (1960) 

"Lateral Earthpressure and Stability of Quay Walls During Earth- 
quake," proc. 2nd World Conference on Earthquake Engineering, 
Tokyo, Japan, Vol. 1, July 1960, pp 165-181. 

70. Marcal, P. V. (1965) 

"A Stiffness Method for Elastic-Plastic Problems," International 
Journal of Mechanics Science, Vol. 7, No. 4, Apr. 1965, pp 229-238. 

71. McCormick, J. M., and Salvadori (1964) 

"Numerical Methods in Fortran," Prentice-Hall, Inc., New Jersey. 

72. Minami, T. (1972) 

"Elasto-Plastic Earthquake Response of Soil-Building System, " 
Ph.D. Thesis, University of California, 1972. 

73. Morris, G. A., and Fenves, S. J. (1967) 

"A General Procedure for the Analysis of Elastic and Plastic Frame- 
works," Structural Research Series No. 325, Civil Engineering 
Studies, University of Illinois, Urbana, Illinois, Aug. 1967. 

74. Mononobe, N., and Matsuo, H. (1929) 

"On the Determination of Earthpressure During Earthquake," proc, 
World Engineering Congress, Tokyo, 1929, Vol. IX, paper no. 388, 
pp 177-186. 

75. Nandakaumaran, P., and Prakash, S. (1970) 

"The Problem of Retaining Walls in Seismic Zones," 4th Symposium on 
Earthquake Engineering, Indian Society of Earthquake Technology, 
Nov. 1970, pp 307-310. 

76. Newmark, N. M. (1959) 

"A Method of Computation for Structural Dynamics," proc. ASCE, 
Vol. 85, No. EM3, 1959. 

77. Nigma N. C. (1970) 

"Yielding in Framed Structures Under Dynamic Loads," Journal of 
the Engineering Mechanics, ASCE, Vol. 96, No. EM5, Oct. 1970, 
pp 687-709. 

78. Niwa, Shin (1960) 

"An Experimental Study of Oscillating Earth Pressure Acting on a 
Quay Wall,™ 2nd World Conference on Earthquake Engineering, Tokyo, 
1960, pp 281-296. 

79. Ohara, S. (1960) 

"Experimental Studies of Seismic Active and Passive Earthpressure," 
proc, 3rd Japan Earthquake Engineering Symposium, Tokyo, Nov. 1960. 



-125- 



80. Okabe, S. (1926) 

"General Theory of Ear thpr essur e , " Journal of Japanese Society of 
Civil Engineering. 

81. Parmelee, R. A. (1967) 

"Building-Foundation Interaction Effect," proc. ASCE, Vol. 93, 
No. EM2, Apr. 1967, pp 131-152. 

82. Penzien, J. (1970) 

"Sqil-Pile Foundation Interaction," "Earthquake Engineering," 
R. L. Wiegel, Coordinating Editor, Prentice-Hall. 

83. Porter, F. L., and Powell, G. H. (1971) 

"Static and Dynamic Analysis of Inelastic Frame Structures," 
EERC 71-3, Earthquake Engineering Research Center, University of 
California, Berkeley, 1971. 

84. Poulos, H. G. (1971) 

"Behavior of Laterally Load Piles: I - Single Pile, II - Pile 
Group," Journal of the Soil Mechanics and Foundation Division, 
ASCE, Vol. 97, No. SM5, proc. paper 8092, May 1971, pp 711-751. 

85. Prager, W. (1955) 

"The Theory of Plasticity: A Survey of Recent Achievements," 

proc. Institute of Mechanical Engineering, Vol. 169, 1955, pp 41-52. 

86. Przemieniecki, J. S. (1968) 

"Theory of Matrix Structural Analysis," McGraw-Hill. 

87. Radhakrishnan, N., and Reese, L^. C. (1970) 

"A Review of Application of Finite Element Method of Analysis to 
Problems in Soil and Rock Mechanics," Soil and Foundation, Vol. X, 
No. 3, The Japanese Society and Foundation Engineering, Sept. 
1970, pp 95-112. 

88. Reddy, A. S., and Valsangkar, A. J. (1970) 

"General Solutions for Laterally Loaded Pile in Elastic-Plastic 
Soil," Soil and Foundation, The Japanese Society of Soil Mechanics 
and Foundation Engineering, Vol. X, No. 3, Sept. 1970, pp 66-80. 

89. Richart, F. E. , Hall, J. R. , and Woods, R. D. (1970) 
"Vibration of Soils and Foundations," Prentice-Hall, Inc. 

90. Richart, F. E., Jr. (1963) 

"Foundation Vibration," Transaction of ASCE, Vol. 127, 1962, Part 
1, pp 863-878. 

91. Richart, C. F. (1958) 

"Elementary Seismology," W. F. Freeman and Co. 

92. Ross, G. A., Seed, H. B., and Migliaccio, R. (1969) 

"Bridge Foundation Behavior in Alaska Earthquake, Journal of Soil ' 

-126- 



Mechanics and Foundation Division, ASCE, July 1969. 

93. Sarrazin, Mauricio A. (1972) 

"Dynamic Soil-Structure Interaction," Journal of the Structural 
Division, ASCE, Vol. 98, No. ST7, proc. paper 9026, July 1972, 
pp 1525-1544. 

94. Scholes, A., and Strover (1971) 

"The Piecewise-Linear Analysis of Two Connecting Structures In- 
cluding the Effects of Clearance at the Connections," International 
Journal for Numerical Methods in Engineering, Vol. 3, No. 10, 
1971, pp 45-52. 

95. 'seed, H. B.,. and Idriss, I. M. (1969) 

"The Influence of Soil Conditions on Ground Motions During Earth- 
quakes," Journal of Soil Mechanics and Foundation Division, ASCE, 
Vol. 95, No. SMI, Jan. 1969. 

96. Seed, H. B. , and Idriss, I. M. (1969) 

"Soil Modulus and Damping Factors for Dynamic Response Analysis," 
EERC 70-10, Earthquake Engineering Research Center, University of 
California, Berkeley, Dec. 1970. 

97. Seed, H. B. , and Whitman, R. V. (1970) 

"Design of Earth Retaining Structures for Dynamic Loads," Specialty 
Conference "Lateral Stress in the Ground and Design of Earth- 
Retaining Structures," Sponsored by Soil Mechanics and Foundation 
Division, ASCE, Ithaca Section, Cornell University, June 22-24, 
1970. 

98. Seed, H. B. (1972) - 

"Dynamic Characteristics of Soil-Structure System," Conference on 
Planning and Design of Tall Building, Lehigh University, Aug. 1972. 

99. Shackel, B. (1968) 

"Damping Characteristics of Soils," proc. 4th Conference, Austra- 
lia Road Research Board, Vol. 4, part 2, pp 1126. 

100. Shepherd, R. , and Charlson, A. W. (1971) 

"Experimental Determination of the Dynamic Properties of a Bridge 
Structure," Bulletin Seismological Society of America, 1971, Vol. 
61(6), pp 1529-1548. 

101. Shepherd, R. , and Sidwell, G. K. (1973) 

"Investigation of the Dynamic Properties of Fine Concrete Bridges," 
4th Australasian Conference on the Mechanics of Structures and 
Materials, University of Queensland Brisbane, August 1973, pp 261- 
268. 

102. Silver, M. , and Seed, H. B. (1969) 

"The Behavior of Sands Under Seismic Loading Conditions," EERC 
69-10, Earthquake Engineering Research Center, University of 
California, Berkeley, Dec. 1969. 

-127- 



103. Taylor, P. W. (1971) 

"The Properties of Soils Under Dynamic Stress Conditions With 
Applications to the Design of Foundation in Seismic Stress," 
Ph.D. Thesis, School of^ Engineering, University of Auckland, 
Nov. 1971. 

104. Tschebotanoff, G. (1949) 

"Large Scale Earthpressure Tests With Model Flexible Bulkheads," 
Bureau of Yards and Docks, Department of the Navy. 

105. Tseng, W. S., and Penzien, J. (1973) 

"Analystic Investigation of the Seismic Response of Long Multiple 
Span Highway Bridge," EERC 73-12, Earthquake* Engineering Research 
Center', University of California, Berkeley, June 1973. 

106. Valera, J. E. (1968) 

"Seismic Interaction of Graular* Soils and Rigid Retaining Struc- 
tures," Ph.D. Thesis, Department of Civil Engineering, University 
of California, Berkeley. 

107.. Wen, R. K. , and Farhoomand, F. (1970) 

"Dynamic Analysis of Inelastic Space Frames," Journal of the Engi- 
neering Mechanics Division, ASCE, Vol. 96, No. EM5, Oct. 1970, 
pp 667-686. 

108. Whitman, R., Roesset, J., and Dobry, R. (1972) 

"Accuracy of Model Superposition for One-Dimensional Soil Ampli- 
tude Analysis," Structural Publication,' No. 351, Department of 
Civil Engineering, School of Engineering, Massachusetts Institute 
of Technology, 1972. 

109. Whitman, R. (1972) 

"Dynamic Soil -Structure Interaction," Structural Publication No. 
352, Department of Civil Engineering, School of Engineering, 
Massachusetts Institute of Technology. 

110. Whitman, R. V., Miller, E.T., and Moore, P. J. (1964) 
"yielding and Locking of Confined Sand , ""Journal of the Soil Mecha- 
nics and Foundation Division, ASCE, Vol. 90, No. SM4, proc. paper 
3966, July 1964, pp 57-84. 

111. White, D. F., and Enderby, L. R. (1970) 

"Finite Element Stress Analysis of a Non-Linear Problem: A 
Connecting-Rod eye by Means of pin," Journal of Strain N Analysis , 
Vol. 5, No. 1, January 1970, pp 41-48. 

112. Wilson, S. D., and Dietrich, R. J. (1960) 

"Effect of Consolidation Pressure on Elastic and Strength Proper- 
ties of Clay," proc, Research Conference on Shear Strength of 
Cohesive Soils, ASCE, Boulder, Colorado, 1960. 



-128- 



113. Wilson, E. L. (1968) 

"A Computer Program for the Dynamic Stress Analysis of Underground 
Structure," Report No. 68-1, SESM, Department of Civil Engineering, 
University of California, Berkeley, Jan. 1968. 

114. Wilson, E. L. (1969) 

"A Method of Analysis for the Evaluation of Foundation-Structure 
Interaction," proc. 4th World Conference on Earthquake Engineering, 
Santiago, Chile, 1969, Vol. 3, pp A 6-87-99. 

115. Wilson, E. L. , and Penzien, J. (1972) 

"Evaluation of Orthogonal Damping Matrices," International Journal 
for Numerical Method in Engineering, Vol. 4, Jan. 1972, pp 5-10. - 

116. Wilson, E. L., and Clough, R. W. (1962) 

"Dynamic Response by Step-by-Step Matrix Analysis," proc, Sympo- 
sium on the Use of Computers in Civil Engineering, Lisbon, Portu- 
gal, 1962. 

117. Wood, J. H. (1973) 

"Earthquake-Induced Soil Pressures on Structures," EERL 70-05, 
Ph.D. Thesis, California Institute of Technology, May 1973. 

118. Yong, R. N. , and McKyes, E. (1971) 

"Yield and Failure of a Clay Under Triaxial Stress," Journal of 
the Soil Mechanics and Foundation Division, ASCE, Vol. 97, No. 
SMI, proc. paper 7790, Jan. 1971, pp 159-176. 

119. Zienkiewicz, 0. C. (1971) 

"The Finite Element Method in Engineering Science," McGraw-Hill, 
Inc. 



-129- 



APPENDIX 



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